Editing Topological vector space

{{Short description|Vector space with a notion of nearness}}

In [[mathematics]], a '''topological vector space''' (also called a '''linear topological space''' and commonly abbreviated '''TVS''' or '''t.v.s.''') is one of the basic structures investigated in [[functional analysis]].
A topological vector space is a [[vector space]] that is also a [[topological space]] with the property that the vector space operations (vector addition and scalar multiplication) are also [[Continuous function|continuous functions]]. Such a topology is called a {{em|vector topology}} and every topological vector space has a [[Uniform space|uniform topological structure]], allowing a notion of [[uniform convergence]] and [[Complete topological vector space|completeness]]. Some authors also require that the space is a [[Hausdorff space]] (although this article does not). One of the most widely studied categories of TVSs are [[locally convex topological vector space]]s. This article focuses on TVSs that are not necessarily locally convex. Other well-known examples of TVSs include [[Banach space]]s, [[Hilbert space]]s and [[Sobolev space]]s.

Many topological vector spaces are spaces of [[Function (mathematics)|function]]s, or [[Linear map|linear operators]] acting on topological vector spaces, and the topology is often defined so as to capture a particular notion of [[Limit (mathematics)#Function space|convergence]] of sequences of functions.

In this article, the [[Scalar multiplication|scalar]] field of a topological vector space will be assumed to be either the [[complex number]]s <math>\Complex</math> or the [[real number]]s <math>\R,</math> unless clearly stated otherwise. 

==Motivation==

===Normed spaces===

Every [[normed vector space]] has a natural [[Normed vector space#Topological structure|topological structure]]: the norm induces a [[Metric space|metric]] and the metric induces a topology.
This is a topological vector space because{{Citation needed|date=April 2024}}:
#The vector addition map <math>\cdot\, + \,\cdot\; : X \times X \to X</math> defined by <math>(x, y) \mapsto x + y</math> is (jointly) continuous with respect to this topology. This follows directly from the [[triangle inequality]] obeyed by the norm.
#The scalar multiplication map <math>\cdot : \mathbb{K} \times X \to X</math> defined by <math>(s, x) \mapsto s \cdot x,</math> where <math>\mathbb{K}</math> is the underlying scalar field of <math>X,</math> is (jointly) continuous. This follows from the triangle inequality and homogeneity of the norm.

Thus all [[Banach space]]s and [[Hilbert space]]s are examples of topological vector spaces.

===Non-normed spaces===

There are topological vector spaces whose topology is not induced by a norm, but are still of interest in analysis. Examples of such spaces are spaces of [[holomorphic function]]s on an open domain, spaces of [[infinitely differentiable function]]s, the [[Schwartz space]]s, and spaces of [[test function]]s and the spaces of [[Distribution (mathematics)|distributions]] on them.{{sfn|Rudin|1991|p=4-5 §1.3}} These are all examples of [[Montel space]]s. An infinite-dimensional Montel space is never normable. The existence of a norm for a given topological vector space is characterized by [[Kolmogorov's normability criterion]]. 

A [[topological field]] is a topological vector space over each of its [[Field extension|subfields]].

==Definition==

[[Image:Topological vector space illust.svg|right|thumb|A family of neighborhoods of the origin with the above two properties determines uniquely a topological vector space. The system of neighborhoods of any other point in the vector space is obtained by [[Translation (geometry)|translation]].]]

A '''topological vector space''' ('''TVS''') <math>X</math> is a [[vector space]] over a [[topological field]] <math>\mathbb{K}</math> (most often the [[Real number|real]] or [[Complex number|complex]] numbers with their standard topologies) that is endowed with a [[Topological space|topology]] such that vector addition <math>\cdot\, + \,\cdot\; : X \times X \to X</math> and scalar multiplication <math>\cdot : \mathbb{K} \times X \to X</math> are [[Continuous function (topology)|continuous functions]] (where the domains of these functions are endowed with [[Product topology|product topologies]]). Such a topology is called a '''{{visible anchor|vector topology}}''' or a '''{{visible anchor|TVS topology}}''' on <math>X.</math>

Every topological vector space is also a commutative [[topological group]] under addition.

'''Hausdorff assumption'''

Many authors (for example, [[Walter Rudin]]), but not this page, require the topology on <math>X</math> to be [[T1 space|T<sub>1</sub>]]; it then follows that the space is [[Hausdorff space|Hausdorff]], and even [[Tychonoff space|Tychonoff]]. A topological vector space is said to be {{em|{{visible anchor|separated}}}} if it is Hausdorff; importantly, "separated" does not mean [[Separable space|separable]]. The topological and linear algebraic structures can be tied together even more closely with additional assumptions, the most common of which are listed [[#Types|below]].

'''Category and morphisms'''

The [[Category (category theory)|category]] of topological vector spaces over a given topological field <math>\mathbb{K}</math> is commonly denoted <math>\mathrm{TVS}_\mathbb{K}</math> or <math>\mathrm{TVect}_\mathbb{K}.</math> The [[Object (category theory)|objects]] are the topological vector spaces over <math>\mathbb{K}</math> and the [[morphism]]s are the [[Continuous linear map|continuous <math>\mathbb{K}</math>-linear map]]s from one object to another.

A {{em|{{visible anchor|topological vector space homomorphism}}}} (abbreviated {{em|{{visible anchor|TVS homomorphism|TVS-homomorphism}}}}), also called a {{em|{{visible anchor|topological homomorphism|text=[[topological homomorphism]]}}}},{{sfn|Köthe|1983|p=91}}{{sfn|Schaefer|Wolff|1999|pp=74–78}} is a [[Continuous map|continuous]] [[linear map]] <math>u : X \to Y</math> between topological vector spaces (TVSs) such that the induced map <math>u : X \to \operatorname{Im} u</math> is an [[open mapping]] when <math>\operatorname{Im} u := u(X),</math> which is the range or image of <math>u,</math> is given the [[subspace topology]] induced by <math>Y.</math>

A {{em|{{visible anchor|topological vector space embedding}}}} (abbreviated {{em|{{visible anchor|TVS embedding|TVS-embedding}}}}), also called a {{em|{{visible anchor|topological monomorphism|text=topological [[monomorphism]]}}}}, is an [[Injective map|injective]] topological homomorphism. Equivalently, a TVS-embedding is a linear map that is also a [[topological embedding]].{{sfn|Köthe|1983|p=91}}

A {{em|{{visible anchor|topological vector space isomorphism}}}} (abbreviated {{em|{{visible anchor|TVS isomorphism|TVS-isomorphism}}}}), also called a {{em|{{visible anchor|topological vector isomorphism}}}}{{sfn|Grothendieck|1973|pp=34-36}} or an {{em|{{visible anchor|isomorphism in the category of TVSs|isomorphism in the category of topological vector spaces}}}}, is a bijective [[Linear map|linear]] [[homeomorphism]]. Equivalently, it is a [[Surjective map|surjective]] TVS embedding{{sfn|Köthe|1983|p=91}}

Many properties of TVSs that are studied, such as [[Locally convex topological vector space|local convexity]], [[Metrizable topological vector space|metrizability]], [[Complete topological vector space|completeness]], and [[Normable space|normability]], are invariant under TVS isomorphisms.

'''A necessary condition for a vector topology'''

A collection <math>\mathcal{N}</math> of subsets of a vector space is called {{em|additive}}{{sfn|Wilansky|2013|pp=40-47}} if for every <math>N \in \mathcal{N},</math> there exists some <math>U \in \mathcal{N}</math> such that <math>U + U \subseteq N.</math>

{{Math theorem|name=Characterization of continuity of addition at <math>0</math>{{sfn|Wilansky|2013|pp=40-47}}|note=|math_statement=
If <math>(X, +)</math> is a [[Group (mathematics)|group]] (as all vector spaces are), <math>\tau</math> is a topology on <math>X,</math> and <math>X \times X</math> is endowed with the [[product topology]], then the addition map <math>X \times X \to X</math> (defined by <math>(x, y) \mapsto x + y</math>) is continuous at the origin of <math>X \times X</math> if and only if the set of [[Neighborhood (topology)|neighborhood]]s of the origin in <math>(X, \tau)</math> is additive. This statement remains true if the word "neighborhood" is replaced by "open neighborhood."
}}

All of the above conditions are consequently a necessity for a topology to form a vector topology.

===Defining topologies using neighborhoods of the origin===

Since every vector topology is translation invariant (which means that for all <math>x_0 \in X,</math> the map <math>X \to X</math> defined by <math>x \mapsto x_0 + x</math> is a [[homeomorphism]]), to define a vector topology it suffices to define a [[neighborhood basis]] (or subbasis) for it at the origin.

{{Math theorem|name=Theorem{{sfn|Narici|Beckenstein|2011|pp=67-113}}|note=Neighborhood filter of the origin|math_statement=
Suppose that <math>X</math> is a real or complex vector space. If <math>\mathcal{B}</math> is a [[Empty set|non-empty]] additive collection of [[Balanced set|balanced]] and [[Absorbing set|absorbing]] subsets of <math>X</math> then <math>\mathcal{B}</math> is a [[neighborhood base]] at <math>0</math> for a vector topology on <math>X.</math> That is, the assumptions are that <math>\mathcal{B}</math> is a [[filter base]] that satisfies the following conditions:
# Every <math>B \in \mathcal{B}</math> is [[Balanced set|balanced]] and [[Absorbing set|absorbing]],
# <math>\mathcal{B}</math> is additive: For every <math>B \in \mathcal{B}</math> there exists a <math>U \in \mathcal{B}</math> such that <math>U + U \subseteq B,</math>

If <math>\mathcal{B}</math> satisfies the above two conditions but is {{em|not}} a filter base then it will form a neighborhood {{em|sub}}basis at <math>0</math> (rather than a neighborhood basis) for a vector topology on <math>X.</math>
}}

In general, the set of all balanced and absorbing subsets of a vector space does not satisfy the conditions of this theorem and does not form a neighborhood basis at the origin for any vector topology.{{sfn|Wilansky|2013|pp=40-47}}

{{anchor|String|Strings}}
===Defining topologies using strings===

Let <math>X</math> be a vector space and let <math>U_{\bull} = \left(U_i\right)_{i = 1}^{\infty}</math> be a sequence of subsets of <math>X.</math> Each set in the sequence <math>U_{\bull}</math> is called a '''{{visible anchor|knot}}''' of <math>U_{\bull}</math> and for every index <math>i,</math> <math>U_i</math> is called the '''<math>i</math>-th knot''' of <math>U_{\bull}.</math> The set <math>U_1</math> is called the '''beginning''' of <math>U_{\bull}.</math> The sequence <math>U_{\bull}</math> is/is a:{{sfn|Adasch|Ernst|Keim|1978|pp=5-9}}{{sfn|Schechter|1996|pp=721-751}}{{sfn|Narici|Beckenstein|2011|pp=371-423}}

* '''{{visible anchor|Summative}}''' if <math>U_{i+1} + U_{i+1} \subseteq U_i</math> for every index <math>i.</math>
* '''[[Balanced set|Balanced]]''' (resp. '''[[Absorbing set|absorbing]]''', '''closed''',<ref group="note">The topological properties of course also require that <math>X</math> be a TVS.</ref> '''convex''', '''open''', '''[[Symmetric set|symmetric]]''', '''[[Barrelled space|barrelled]]''', '''[[Absolutely convex set|absolutely convex/disked]]''', etc.) if this is true of every <math>U_i.</math>
* '''{{visible anchor|String}}''' if <math>U_{\bull}</math> is summative, absorbing, and balanced.
* '''{{visible anchor|Topological string}}''' or a '''{{visible anchor|neighborhood string}}''' in a TVS <math>X</math> if <math>U_{\bull}</math> is a string and each of its knots is a neighborhood of the origin in <math>X.</math>

<!-------- START: REMOVED DEFINTION --------------
'''Definition''' ('''Ultrabarrel'''/'''suprabarrel'''): A subset of a TVS <math>X</math> is called an '''ultrabarrel''' (resp. '''suprabarrel''') if it is the beginning of some closed string (resp. of some string) in <math>X.</math>
----------- END: REMOVED DEFINTION --------------->
If <math>U</math> is an [[Absorbing set|absorbing]] [[Absolutely convex set|disk]] in a vector space <math>X</math> then the sequence defined by <math>U_i := 2^{1-i} U</math> forms a string beginning with <math>U_1 = U.</math> This is called the '''natural string of <math>U</math>'''{{sfn|Adasch|Ernst|Keim|1978|pp=5-9}} Moreover, if a vector space <math>X</math> has countable dimension then every string contains an [[Absolutely convex set|absolutely convex]] string.

Summative sequences of sets have the particularly nice property that they define non-negative continuous real-valued [[subadditive]] functions. These functions can then be used to prove many of the basic properties of topological vector spaces.

{{Math theorem|name=Theorem|note=<math>\R</math>-valued function induced by a string|math_statement=
Let <math>U_{\bull} = \left(U_i\right)_{i=0}^{\infty}</math> be a collection of subsets of a vector space such that <math>0 \in U_i</math> and <math>U_{i+1} + U_{i+1} \subseteq U_i</math> for all <math>i \geq 0.</math> For all <math>u \in U_0,</math> let <math display=block>\mathbb{S}(u) := \left\{n_{\bull} = \left(n_1, \ldots, n_k\right) ~:~ k \geq 1, n_i \geq 0 \text{ for all } i, \text{ and } u \in U_{n_1} + \cdots + U_{n_k}\right\}.</math>

Define <math>f : X \to [0, 1]</math> by <math>f(x) = 1</math> if <math>x \not\in U_0</math> and otherwise let <math display=block>f(x) := \inf_{} \left\{2^{- n_1} + \cdots 2^{- n_k} ~:~ n_{\bull} = \left(n_1, \ldots, n_k\right) \in \mathbb{S}(x)\right\}.</math>

Then <math>f</math> is subadditive (meaning <math>f(x + y) \leq f(x) + f(y)</math> for all <math>x, y \in X</math>) and <math>f = 0</math> on <math display=inline>\bigcap_{i \geq 0} U_i;</math> so in particular, <math>f(0) = 0.</math> If all <math>U_i</math> are [[symmetric set]]s then <math>f(-x) = f(x)</math> and if all <math>U_i</math> are balanced then <math>f(s x) \leq f(x)</math> for all scalars <math>s</math> such that <math>|s| \leq 1</math> and all <math>x \in X.</math> If <math>X</math> is a topological vector space and if all <math>U_i</math> are neighborhoods of the origin then <math>f</math> is continuous, where if in addition <math>X</math> is Hausdorff and <math>U_{\bull}</math> forms a basis of balanced neighborhoods of the origin in <math>X</math> then <math>d(x, y) := f(x - y)</math> is a metric defining the vector topology on <math>X.</math>
<!--- This theorem is true more generally for commutative additive [[topological group]]s. --->
}}
A proof of the above theorem is given in the article on [[Metrizable topological vector space#Additive sequences|metrizable topological vector spaces]].

If <math>U_{\bull} = \left(U_i\right)_{i \in \N}</math> and <math>V_{\bull} = \left(V_i\right)_{i \in \N}</math> are two collections of subsets of a vector space <math>X</math> and if <math>s</math> is a scalar, then by definition:{{sfn|Adasch|Ernst|Keim|1978|pp=5-9}}

* <math>V_{\bull}</math> '''contains''' <math>U_{\bull}</math>: <math>\ U_{\bull} \subseteq V_{\bull}</math> if and only if <math>U_i \subseteq V_i</math> for every index <math>i.</math>
* '''Set of knots''': <math>\ \operatorname{Knots} U_{\bull} := \left\{U_i : i \in \N\right\}.</math>
* '''Kernel''': <math display=inline>\ \ker U_{\bull} := \bigcap_{i \in \N} U_i.</math>
* '''Scalar multiple''': <math>\ s U_{\bull} := \left(s U_i\right)_{i \in \N}.</math>
* '''Sum''': <math>\ U_{\bull} + V_{\bull} := \left(U_i + V_i\right)_{i \in \N}.</math>
* '''Intersection''': <math>\ U_{\bull} \cap V_{\bull} := \left(U_i \cap V_i\right)_{i \in \N}.</math>

If <math>\mathbb{S}</math> is a collection sequences of subsets of <math>X,</math> then <math>\mathbb{S}</math> is said to be '''directed''' ('''downwards''') '''under inclusion''' or simply '''directed downward''' if <math>\mathbb{S}</math> is not empty and for all <math>U_{\bull}, V_{\bull} \in \mathbb{S},</math> there exists some <math>W_{\bull} \in \mathbb{S}</math> such that <math>W_{\bull} \subseteq U_{\bull}</math> and <math>W_{\bull} \subseteq V_{\bull}</math> (said differently, if and only if <math>\mathbb{S}</math> is a [[Filter (set theory)|prefilter]] with respect to the containment <math>\,\subseteq\,</math> defined above).

'''Notation''': Let <math display=inline>\operatorname{Knots} \mathbb{S} := \bigcup_{U_{\bull} \in \mathbb{S}} \operatorname{Knots} U_{\bull}</math> be the set of all knots of all strings in <math>\mathbb{S}.</math>

Defining vector topologies using collections of strings is particularly useful for defining classes of TVSs that are not necessarily locally convex.

{{Math theorem|name=Theorem{{sfn|Adasch|Ernst|Keim|1978|pp=5-9}}|note=Topology induced by strings|math_statement=If <math>(X, \tau)</math> is a topological vector space then there exists a set <math>\mathbb{S}</math><ref group=proof>This condition is satisfied if <math>\mathbb{S}</math> denotes the set of all topological strings in <math>(X, \tau).</math></ref> of neighborhood strings in <math>X</math> that is directed downward and such that the set of all knots of all strings in <math>\mathbb{S}</math> is a [[neighborhood basis]] at the origin for <math>(X, \tau).</math> Such a collection of strings is said to be {{em|<math>\tau</math> '''fundamental'''}}.

Conversely, if <math>X</math> is a vector space and if <math>\mathbb{S}</math> is a collection of strings in <math>X</math> that is directed downward, then the set <math>\operatorname{Knots} \mathbb{S}</math> of all knots of all strings in <math>\mathbb{S}</math> forms a [[neighborhood basis]] at the origin for a vector topology on <math>X.</math> In this case, this topology is denoted by <math>\tau_\mathbb{S}</math> and it is called the '''topology generated by <math>\mathbb{S}.</math>'''
}}

If <math>\mathbb{S}</math> is the set of all topological strings in a TVS <math>(X, \tau)</math> then <math>\tau_{\mathbb{S}} = \tau.</math>{{sfn|Adasch|Ernst|Keim|1978|pp=5-9}} A Hausdorff TVS is [[Metrizable topological vector space|metrizable]] [[if and only if]] its topology can be induced by a single topological string.{{sfn|Adasch|Ernst|Keim|1978|pp=10-15}}

==Topological structure==

A vector space is an [[abelian group]] with respect to the operation of addition, and in a topological vector space the inverse operation is always continuous (since it is the same as multiplication by <math>-1</math>). Hence, every topological vector space is an abelian [[topological group]]. Every TVS is [[completely regular]] but a TVS need not be [[Normal space|normal]].{{sfn|Wilansky|2013|p=53}}

Let <math>X</math> be a topological vector space. Given a [[Subspace topology|subspace]] <math>M \subseteq X,</math> the quotient space <math>X / M</math> with the usual [[quotient space (topology)|quotient topology]] is a Hausdorff topological vector space if and only if <math>M</math> is closed.<ref group=note>In particular, <math>X</math> is Hausdorff if and only if the set <math>\{0\}</math> is closed (that is, <math>X</math> is a [[T1 space|T<sub>1</sub> space]]).</ref> This permits the following construction: given a topological vector space <math>X</math> (that is probably not Hausdorff), form the quotient space <math>X / M</math> where <math>M</math> is the closure of <math>\{0\}.</math> <math>X / M</math> is then a Hausdorff topological vector space that can be studied instead of <math>X.</math>

===Invariance of vector topologies===

One of the most used properties of vector topologies is that every vector topology is {{em|{{visible anchor|translation invariant}}}}:
:for all <math>x_0 \in X,</math> the map <math>X \to X</math> defined by <math>x \mapsto x_0 + x</math> is a [[homeomorphism]], but if <math>x_0 \neq 0</math> then it is not linear and so not a TVS-isomorphism.
Scalar multiplication by a non-zero scalar is a TVS-isomorphism. This means that if <math>s \neq 0</math> then the linear map <math>X \to X</math> defined by <math>x \mapsto s x</math> is a homeomorphism. Using <math>s = -1</math> produces the negation map <math>X \to X</math> defined by <math>x \mapsto - x,</math> which is consequently a linear homeomorphism and thus a TVS-isomorphism.

If <math>x \in X</math> and any subset <math>S \subseteq X,</math> then <math>\operatorname{cl}_X (x + S) = x + \operatorname{cl}_X S</math>{{sfn|Narici|Beckenstein|2011|pp=67-113}} and moreover, if <math>0 \in S</math> then <math>x + S</math> is a [[Neighborhood (topology)|neighborhood]] (resp. open neighborhood, closed neighborhood) of <math>x</math> in <math>X</math> if and only if the same is true of <math>S</math> at the origin.

===Local notions===

A subset <math>E</math> of a vector space <math>X</math> is said to be
* '''[[Absorbing set|absorbing]]''' (in <math>X</math>): if for every <math>x \in X,</math> there exists a real <math>r > 0</math> such that <math>c x \in E</math> for any scalar <math>c</math> satisfying <math>|c| \leq r.</math>{{sfn|Rudin|1991|p=6 §1.4}}
* '''[[Balanced set|balanced]]''' or '''circled''': if <math>t E \subseteq E</math> for every scalar <math>|t| \leq 1.</math>{{sfn|Rudin|1991|p=6 §1.4}}
* '''[[Convex set|convex]]''': if <math>t E + (1 - t) E \subseteq E</math> for every real <math>0 \leq t \leq 1.</math>{{sfn|Rudin|1991|p=6 §1.4}}
* a '''[[Absolutely convex set|disk]]''' or '''[[Absolutely convex set|absolutely convex]]''': if <math>E</math> is convex and balanced.
* '''[[Symmetric set|symmetric]]''': if <math>- E \subseteq E,</math> or equivalently, if <math>- E = E.</math>

Every neighborhood of the origin is an [[absorbing set]] and contains an open [[Balanced set|balanced]] neighborhood of <math>0</math>{{sfn|Narici|Beckenstein|2011|pp=67-113}} so every topological vector space has a local base of absorbing and [[balanced set]]s. The origin even has a neighborhood basis consisting of closed balanced neighborhoods of <math>0;</math> if the space is [[locally convex]] then it also has a neighborhood basis consisting of closed convex balanced neighborhoods of the origin.

'''Bounded subsets'''

A subset <math>E</math> of a topological vector space <math>X</math> is '''[[Bounded set (topological vector space)|bounded]]'''{{sfn|Rudin|1991|p=8}} if for every neighborhood <math>V</math> of the origin there exists <math>t</math> such that <math>E \subseteq t V</math>.

The definition of boundedness can be weakened a bit; <math>E</math> is bounded if and only if every countable subset of it is bounded. A set is bounded if and only if each of its subsequences is a bounded set.{{sfn|Narici|Beckenstein|2011|pp=155-176}} Also, <math>E</math> is bounded if and only if for every balanced neighborhood <math>V</math> of the origin, there exists <math>t</math> such that <math>E \subseteq t V.</math> Moreover, when <math>X</math> is locally convex, the boundedness can be characterized by [[seminorm]]s: the subset <math>E</math> is bounded if and only if every continuous seminorm <math>p</math> is bounded on <math>E.</math>{{sfn|Rudin|1991|p=27-28 Theorem 1.37}}

Every [[totally bounded]] set is bounded.{{sfn|Narici|Beckenstein|2011|pp=155-176}} If <math>M</math> is a vector subspace of a TVS <math>X,</math> then a subset of <math>M</math> is bounded in <math>M</math> if and only if it is bounded in <math>X.</math>{{sfn|Narici|Beckenstein|2011|pp=155-176}}

===Metrizability===

{{Math theorem|name=[[Birkhoff–Kakutani theorem]]|math_statement=
If <math>(X, \tau)</math> is a topological vector space then the following four conditions are equivalent:{{sfn|Köthe|1983|loc=section 15.11}}<ref group=note>In fact, this is true for topological group, since the proof does not use the scalar multiplications.</ref>
# The origin <math>\{0\}</math> is closed in <math>X</math> and there is a [[countable]] [[neighborhood basis|basis of neighborhoods]] at the origin in <math>X.</math>
# <math>(X, \tau)</math> is [[Metrizable space|metrizable]] (as a topological space).
# There is a [[translation-invariant metric]] on <math>X</math> that induces on <math>X</math> the topology <math>\tau,</math> which is the given topology on <math>X.</math>
# <math>(X, \tau)</math> is a [[metrizable topological vector space]].<ref group=note>Also called a '''metric linear space''', which means that it is a real or complex vector space together with a translation-invariant metric for which addition and scalar multiplication are continuous.</ref>

By the Birkhoff–Kakutani theorem, it follows that there is an [[Equivalence of metrics|equivalent metric]] that is translation-invariant.
}}

A TVS is [[Metrizable TVS|pseudometrizable]] if and only if it has a countable neighborhood basis at the origin, or equivalent, if and only if its topology is generated by an [[Metrizable TVS|''F''-seminorm]]. A TVS is metrizable if and only if it is Hausdorff and pseudometrizable.

More strongly: a topological vector space is said to be '''[[normable]]''' if its topology can be induced by a norm. A topological vector space is normable if and only if it is Hausdorff and has a convex bounded neighborhood of the origin.<ref name="springer">{{SpringerEOM|title=Topological vector space|access-date=26 February 2021}}</ref>

Let <math>\mathbb{K}</math> be a non-[[Discrete space|discrete]] [[locally compact]] topological field, for example the real or complex numbers. A [[Hausdorff space|Hausdorff]] topological vector space over <math>\mathbb{K}</math> is locally compact if and only if it is [[finite-dimensional]], that is, isomorphic to <math>\mathbb{K}^n</math> for some natural number <math>n.</math>{{sfn|Rudin|1991|p=17 Theorem 1.22}}

===Completeness and uniform structure===
{{Main|Complete topological vector space}}

The '''[[Complete topological vector space|canonical uniformity]]'''{{sfn|Schaefer|Wolff|1999|pp=12-19}} on a TVS <math>(X, \tau)</math> is the unique translation-invariant [[Uniform space|uniformity]] that induces the topology <math>\tau</math> on <math>X.</math>

Every TVS is assumed to be endowed with this canonical uniformity, which makes all TVSs into [[uniform space]]s. This allows one to talk{{clarify|date=September 2020}} about related notions such as [[Complete topological vector space|completeness]], [[uniform convergence]], Cauchy nets, and [[uniform continuity]], etc., which are always assumed to be with respect to this uniformity (unless indicated other). This implies that every Hausdorff topological vector space is [[Tychonoff space|Tychonoff]].{{sfn|Schaefer|Wolff|1999|p=16}} A subset of a TVS is [[Compact space|compact]] if and only if it is complete and [[totally bounded]] (for Hausdorff TVSs, a set being totally bounded is equivalent to it being [[Totally bounded space#In topological groups|precompact]]). But if the TVS is not Hausdorff then there exist compact subsets that are not closed. However, the closure of a compact subset of a non-Hausdorff TVS is again compact (so compact subsets are [[relatively compact]]).

With respect to this uniformity, a [[Net (mathematics)|net]] (or [[Sequence (mathematics)|sequence]]) <math>x_{\bull} = \left(x_i\right)_{i \in I}</math> is '''Cauchy''' if and only if for every neighborhood <math>V</math> of <math>0,</math> there exists some index <math>n</math> such that <math>x_i - x_j \in V</math> whenever <math>i \geq n</math> and <math>j \geq n.</math>

Every [[Cauchy sequence]] is bounded, although Cauchy nets and Cauchy filters may not be bounded. A topological vector space where every Cauchy sequence converges is called '''[[sequentially complete]]'''; in general, it may not be complete (in the sense that all Cauchy filters converge).

The vector space operation of addition is uniformly continuous and an [[Open and closed map|open map]]. Scalar multiplication is [[Cauchy continuous]] but in general, it is almost never uniformly continuous. Because of this, every topological vector space can be completed and is thus a [[Dense set|dense]] [[linear subspace]] of a [[complete topological vector space]].

* Every TVS has a [[Complete topological vector space|completion]] and every Hausdorff TVS has a Hausdorff completion.{{sfn|Narici|Beckenstein|2011|pp=67-113}} Every TVS (even those that are Hausdorff and/or complete) has infinitely many non-isomorphic non-Hausdorff completions.
* A compact subset of a TVS (not necessarily Hausdorff) is complete.{{sfn|Narici|Beckenstein|2011|pp=115-154}} A complete subset of a Hausdorff TVS is closed.{{sfn|Narici|Beckenstein|2011|pp=115-154}}
* If <math>C</math> is a complete subset of a TVS then any subset of <math>C</math> that is closed in <math>C</math> is complete.{{sfn|Narici|Beckenstein|2011|pp=115-154}}
* A Cauchy sequence in a Hausdorff TVS <math>X</math> is not necessarily [[relatively compact]] (that is, its closure in <math>X</math> is not necessarily compact).
* If a Cauchy filter in a TVS has an [[Filters in topology|accumulation point]] <math>x</math> then it converges to <math>x.</math>
* If a series <math display=inline>\sum_{i=1}^{\infty} x_i</math> converges<ref group="note">A series <math display=inline>\sum_{i=1}^{\infty} x_i</math> is said to '''converge''' in a TVS <math>X</math> if the sequence of partial sums converges.</ref> in a TVS <math>X</math> then <math>x_{\bull} \to 0</math> in <math>X.</math>{{sfn|Swartz|1992|pp=27-29}}

==Examples==

===Finest and coarsest vector topology===

Let <math>X</math> be a real or complex vector space.

'''Trivial topology'''

The '''[[trivial topology]]''' or '''indiscrete topology''' <math>\{X, \varnothing\}</math> is always a TVS topology on any vector space <math>X</math> and it is the coarsest TVS topology possible. An important consequence of this is that the intersection of any collection of TVS topologies on <math>X</math> always contains a TVS topology. Any vector space (including those that are infinite dimensional) endowed with the trivial topology is a compact (and thus [[locally compact]]) [[Complete topological vector space|complete]] [[Metrizable topological vector space|pseudometrizable]] [[Seminormed space|seminormable]] [[Locally convex topological vector space|locally convex]] topological vector space. It is [[Hausdorff space|Hausdorff]] if and only if <math>\dim X = 0.</math>

'''Finest vector topology'''

There exists a TVS topology <math>\tau_f</math> on <math>X,</math> called the '''{{visible anchor|finest vector topology}}''' on <math>X,</math> that is finer than every other TVS-topology on <math>X</math> (that is, any TVS-topology on <math>X</math> is necessarily a subset of <math>\tau_f</math>).<ref>{{Cite web|date=2016-04-22|title=A quick application of the closed graph theorem|url=https://terrytao.wordpress.com/2016/04/22/a-quick-application-of-the-closed-graph-theorem/|access-date=2020-10-07| website=What's new| language=en}}</ref>{{sfn|Narici|Beckenstein|2011|p=111}} Every linear map from <math>\left(X, \tau_f\right)</math> into another TVS is necessarily continuous. If <math>X</math> has an uncountable [[Hamel basis]] then <math>\tau_f</math> is {{em|not}} [[Locally convex topological vector space|locally convex]] and {{em|not}} [[Metrizable topological vector space|metrizable]].{{sfn|Narici|Beckenstein|2011|p=111}}

===Cartesian products===

A [[Cartesian product]] of a family of topological vector spaces, when endowed with the [[product topology]], is a topological vector space. Consider for instance the set <math>X</math> of all functions <math>f: \R \to \R</math> where <math>\R</math> carries its usual [[Euclidean topology]]. This set <math>X</math> is a real vector space (where addition and scalar multiplication are defined pointwise, as usual) that can be identified with (and indeed, is often defined to be) the [[Cartesian product]] <math>\R^\R,,</math> which carries the natural [[product topology]]. With this product topology, <math>X := \R^{\R}</math> becomes a topological vector space whose topology is called {{em|the topology of [[pointwise convergence]] on <math>\R.</math>}} The reason for this name is the following: if <math>\left(f_n\right)_{n=1}^{\infty}</math> is a [[sequence]] (or more generally, a [[Net (mathematics)|net]]) of elements in <math>X</math> and if <math>f \in X</math> then <math>f_n</math> [[limit of a sequence|converges]] to <math>f</math> in <math>X</math> if and only if for every real number <math>x,</math> <math>f_n(x)</math> converges to <math>f(x)</math> in <math>\R.</math>  This TVS is [[Complete topological vector space|complete]], [[Hausdorff space|Hausdorff]], and [[locally convex]] but not [[Metrizable topological vector space|metrizable]] and consequently not [[normable]]; indeed, every neighborhood of the origin in the product topology contains lines (that is, 1-dimensional vector subspaces, which are subsets of the form <math>\R f := \{r f : r \in \R\}</math> with <math>f \neq 0</math>).

===Finite-dimensional spaces===

By [[F. Riesz's theorem]], a Hausdorff topological vector space is finite-dimensional if and only if it is [[locally compact]], which happens if and only if it has a compact [[Neighborhood (topology)|neighborhood]] of the origin.

Let <math>\mathbb{K}</math> denote <math>\R</math> or <math>\Complex</math> and endow <math>\mathbb{K}</math> with its usual Hausdorff normed [[Euclidean topology]].  Let <math>X</math> be a vector space over <math>\mathbb{K}</math> of finite dimension <math>n := \dim X</math> and so that <math>X</math> is vector space isomorphic to <math>\mathbb{K}^n</math> (explicitly, this means that there exists a [[linear isomorphism]] between the vector spaces <math>X</math> and <math>\mathbb{K}^n</math>). This finite-dimensional vector space <math>X</math> always has a unique {{em|[[Hausdorff space|Hausdorff]]}} vector topology, which makes it TVS-isomorphic to <math>\mathbb{K}^n,</math> where <math>\mathbb{K}^n</math> is endowed with the usual Euclidean topology (which is the same as the [[product topology]]). This Hausdorff vector topology is also the (unique) [[Comparison of topologies|finest]] vector topology on <math>X.</math>  <math>X</math> has a unique vector topology if and only if <math>\dim X = 0.</math> If <math>\dim X \neq 0</math> then although <math>X</math> does not have a unique vector topology, it does have a unique {{em|Hausdorff}} vector topology.

* If <math>\dim X = 0</math> then <math>X = \{0\}</math> has exactly one vector topology: the [[trivial topology]], which in this case (and {{em|only}} in this case) is Hausdorff. The trivial topology on a vector space is Hausdorff if and only if the vector space has dimension <math>0.</math>
* If <math>\dim X = 1</math> then <math>X</math> has two vector topologies: the usual [[Euclidean topology]] and the (non-Hausdorff) trivial topology.
** Since the field <math>\mathbb{K}</math> is itself a <math>1</math>-dimensional topological vector space over <math>\mathbb{K}</math> and since it plays an important role in the definition of topological vector spaces, this dichotomy plays an important role in the definition of an [[absorbing set]] and has consequences that reverberate throughout [[functional analysis]].
{{math proof | title=Proof outline| proof =
The proof of this dichotomy (i.e. that a vector topology is either trivial or isomorphic to <math>\mathbb{K}</math>) is straightforward so only an outline with the important observations is given. As usual, <math>\mathbb{K}</math> is assumed have the (normed) Euclidean topology. Let <math>B_r := \{a \in \mathbb{K} : |a| < r\}</math> for all <math>r > 0.</math> Let <math>X</math> be a <math>1</math>-dimensional vector space over <math>\mathbb{K}.</math> If <math>S \subseteq X</math> and <math>B \subseteq \mathbb{K}</math> is a ball centered at <math>0</math> then <math>B \cdot S = X</math> whenever <math>S</math> contains an "unbounded sequence", by which it is meant a sequence of the form <math>\left(a_i x\right)_{i=1}^{\infty}</math> where <math>0 \neq x \in X</math> and <math>\left(a_i\right)_{i=1}^{\infty} \subseteq \mathbb{K}</math> is unbounded in normed space <math>\mathbb{K}</math> (in the usual sense). Any vector topology on <math>X</math> will be translation invariant and invariant under non-zero scalar multiplication, and for every <math>0 \neq x \in X,</math> the map <math>M_x : \mathbb{K} \to X</math> given by <math>M_x(a) := a x</math> is a continuous linear bijection. Because <math>X = \mathbb{K} x</math> for any such <math>x,</math> every subset of <math>X</math> can be written as <math>F x = M_x(F)</math> for some unique subset <math>F \subseteq \mathbb{K}.</math> And if this vector topology on <math>X</math> has a neighborhood <math>W</math> of the origin that is not equal to all of <math>X,</math> then the continuity of scalar multiplication <math>\mathbb{K} \times X \to X</math> at the origin guarantees the existence of an open ball <math>B_r \subseteq \mathbb{K}</math> centered at <math>0</math> and an open neighborhood <math>S</math> of the origin in <math>X</math> such that <math>B_r \cdot S \subseteq W \neq X,</math> which implies that <math>S</math> does {{em|not}} contain any "unbounded sequence". This implies that for every <math>0 \neq x \in X,</math> there exists some positive integer <math>n</math> such that <math>S \subseteq B_n x.</math> From this, it can be deduced that if <math>X</math> does not carry the trivial topology and if <math>0 \neq x \in X,</math> then for any ball <math>B \subseteq \mathbb{K}</math> center at 0 in <math>\mathbb{K},</math> <math>M_x(B) = B x</math> contains an open neighborhood of the origin in <math>X,</math> which then proves that <math>M_x</math> is a linear [[homeomorphism]]. [[Q.E.D.]] <math>\blacksquare</math>
}}
* If <math>\dim X = n \geq 2</math> then <math>X</math> has {{em|infinitely many}} distinct vector topologies:
** Some of these topologies are now described: Every linear functional <math>f</math> on <math>X,</math> which is vector space isomorphic to <math>\mathbb{K}^n,</math> induces a [[seminorm]] <math>|f| : X \to \R</math> defined by <math>|f|(x) = |f(x)|</math> where <math>\ker f = \ker |f|.</math> Every seminorm induces a ([[Metrizable TVS|pseudometrizable]] [[locally convex]]) vector topology on <math>X</math> and seminorms with distinct kernels induce distinct topologies so that in particular, seminorms on <math>X</math> that are induced by linear functionals with distinct kernels will induce distinct vector topologies on <math>X.</math>
** However, while there are infinitely many vector topologies on <math>X</math> when <math>\dim X \geq 2,</math> there are, {{em|up to TVS-isomorphism}}, only <math>1 + \dim X</math> vector topologies on <math>X.</math> For instance, if <math>n := \dim X = 2</math> then the vector topologies on <math>X</math> consist of the trivial topology, the Hausdorff Euclidean topology, and then the infinitely many remaining non-trivial non-Euclidean vector topologies on <math>X</math> are all TVS-isomorphic to one another.

===Non-vector topologies===

'''Discrete and cofinite topologies'''

If <math>X</math> is a non-trivial vector space (that is, of non-zero dimension) then the [[discrete topology]] on <math>X</math> (which is always [[Metrizable space|metrizable]]) is {{em|not}} a TVS topology because despite making addition and negation continuous (which makes it into a [[topological group]] under addition), it fails to make scalar multiplication continuous. The [[cofinite topology]] on <math>X</math> (where a subset is open if and only if its complement is finite) is also {{em|not}} a TVS topology on <math>X.</math>

==Linear maps==

A linear operator between two topological vector spaces which is continuous at one point is continuous on the whole domain. Moreover, a linear operator <math>f</math> is continuous if <math>f(X)</math> is bounded (as defined below) for some neighborhood <math>X</math> of the origin.

A [[hyperplane]] in a topological vector space <math>X</math> is either dense or closed. A [[linear functional]] <math>f</math> on a topological vector space <math>X</math> has either dense or closed kernel. Moreover, <math>f</math> is continuous if and only if its [[Kernel (algebra)|kernel]] is [[closed set|closed]].

==Types==

Depending on the application additional constraints are usually enforced on the topological structure of the space. In fact, several principal results in functional analysis fail to hold in general for topological vector spaces: the [[closed graph theorem]], the [[Open mapping theorem (functional analysis)|open mapping theorem]], and the fact that the dual space of the space separates points in the space.

Below are some common topological vector spaces, roughly in order of increasing "niceness."

* [[F-space]]s are [[complete space|complete]] topological vector spaces with a translation-invariant metric.{{sfn|Rudin|1991|p=9 §1.8}} These include [[Lp space|<math>L^p</math> spaces]] for all <math>p > 0.</math>
* [[Locally convex topological vector space]]s: here each point has a [[local base]] consisting of [[convex set]]s.{{sfn|Rudin|1991|p=9 §1.8}} By a technique known as [[Minkowski functional]]s it can be shown that a space is locally convex if and only if its topology can be defined by a family of seminorms.{{sfn|Rudin|1991|p=27 Theorem 1.36}} Local convexity is the minimum requirement for "geometrical" arguments like the [[Hahn–Banach theorem]]. The <math>L^p</math> spaces are locally convex (in fact, Banach spaces) for all <math>p \geq 1,</math> but not for <math>0 < p < 1.</math>
* [[Barrelled space]]s: locally convex spaces where the [[Banach–Steinhaus theorem]] holds.
* [[Bornological space]]: a locally convex space where the [[continuous linear operator]]s to any locally convex space are exactly the [[bounded linear operator]]s.
* [[Stereotype space]]: a locally convex space satisfying a variant of [[reflexive space|reflexivity condition]], where the dual space is endowed with the topology of uniform convergence on [[totally bounded space|totally bounded sets]].
* [[Montel space]]: a barrelled space where every [[closed set|closed]] and [[Bounded set (topological vector space)|bounded set]] is [[compact set|compact]]
* [[Fréchet space]]s: these are complete locally convex spaces where the topology comes from a translation-invariant metric, or equivalently: from a countable family of seminorms. Many interesting spaces of functions fall into this class -- <math>C^\infty(\R)</math> is a Fréchet space under the seminorms <math display=inline>\|f\|_{k,\ell} = \sup_{x\in[-k,k]} |f^{(\ell)}(x)|.</math> A locally convex F-space is a Fréchet space.{{sfn|Rudin|1991|p=9 §1.8}}
* [[LF-space]]s are [[limit (category theory)|limits]] of [[Fréchet space]]s. [[ILH space]]s are [[inverse limit]]s of Hilbert spaces.
* [[Nuclear space]]s: these are locally convex spaces with the property that every bounded map from the nuclear space to an arbitrary Banach space is a [[nuclear operator]].
* [[Normed space]]s and [[seminormed space]]s: locally convex spaces where the topology can be described by a single [[norm (mathematics)|norm]] or [[seminorm (mathematics)|seminorm]]. In normed spaces a linear operator is continuous if and only if it is bounded.
* [[Banach space]]s: Complete [[normed vector space]]s. Most of functional analysis is formulated for Banach spaces. This class includes the <math>L^p</math> spaces with <math>1\leq p \leq \infty,</math> the space <math>BV</math> of [[Bounded variation|functions of bounded variation]], and [[Ba space|certain spaces]] of measures.
* [[Reflexive space|Reflexive Banach space]]s: Banach spaces naturally isomorphic to their double dual (see below), which ensures that some geometrical arguments can be carried out. An important example which is {{em|not}} reflexive is [[Lp space|<math>L^1</math>]], whose dual is <math>L^{\infty}</math> but is strictly contained in the dual of <math>L^{\infty}.</math>
* [[Hilbert space]]s: these have an [[inner product]]; even though these spaces may be infinite-dimensional, most geometrical reasoning familiar from finite dimensions can be carried out in them. These include <math>L^2</math> spaces, the <math>L^2</math> [[Sobolev space|Sobolev spaces]] <math>W^{2,k},</math> and [[Hardy space|Hardy spaces]].
* [[Euclidean space]]s: <math>\R^n</math> or <math>\Complex^n</math> with the topology induced by the standard inner product. As pointed out in the preceding section, for a given finite <math>n,</math> there is only one <math>n</math>-dimensional topological vector space, up to isomorphism. It follows from this that any finite-dimensional subspace of a TVS is closed. A characterization of finite dimensionality is that a Hausdorff TVS is locally compact if and only if it is finite-dimensional (therefore isomorphic to some Euclidean space).

==Dual space==

{{Main|Algebraic dual space|Continuous dual space|Strong dual space}}

Every topological vector space has a [[continuous dual space]]&mdash;the set <math>X'</math> of all continuous linear functionals, that is, [[continuous linear map]]s from the space into the base field <math>\mathbb{K}.</math> A topology on the dual can be defined to be the coarsest topology such that the dual pairing each point evaluation <math>X' \to \mathbb{K}</math> is continuous. This turns the dual into a locally convex topological vector space. This topology is called the [[Weak topology|weak-* topology]].{{sfn|Rudin|1991|p=62-68 §3.8-3.14}} This may not be the only [[natural topology]] on the dual space; for instance, the dual of a normed space has a natural norm defined on it. However, it is very important in applications because of its compactness properties (see [[Banach–Alaoglu theorem]]). Caution: Whenever <math>X</math> is a non-normable locally convex space, then the pairing map <math>X' \times X \to \mathbb{K}</math> is never continuous, no matter which vector space topology one chooses on <math>X'.</math> A topological vector space has a non-trivial continuous dual space if and only if it has a proper convex neighborhood of the origin.{{sfn|Narici|Beckenstein|2011|pp=177-220}}

==Properties==
{{See also|Locally convex topological vector space#Properties}}

For any <math>S \subseteq X</math> of a TVS <math>X,</math> the [[Convex set|''convex'']] (resp. ''[[Balanced set|balanced]], [[Absolutely convex set|disked]], closed convex, closed balanced, closed disked''') ''hull'' of <math>S</math> is the smallest subset of <math>X</math> that has this property and contains <math>S.</math> The closure (respectively, interior, [[convex hull]], balanced hull, disked hull) of a set <math>S</math> is sometimes denoted by <math>\operatorname{cl}_X S</math> (respectively, <math>\operatorname{Int}_X S,</math> <math>\operatorname{co} S,</math> <math>\operatorname{bal} S,</math> <math>\operatorname{cobal} S</math>).

The [[convex hull]] <math>\operatorname{co} S</math> of a subset <math>S</math> is equal to the set of all {{em|[[convex combination]]s}} of elements in <math>S,</math> which are finite [[linear combination]]s of the form <math>t_1 s_1 + \cdots + t_n s_n</math> where <math>n \geq 1</math> is an integer, <math>s_1, \ldots, s_n \in S</math> and <math>t_1, \ldots, t_n \in [0, 1]</math> sum to <math>1.</math>{{sfn|Rudin|1991|p=38}} The intersection of any family of convex sets is convex and the convex hull of a subset is equal to the intersection of all convex sets that contain it.{{sfn|Rudin|1991|p=38}}

===Neighborhoods and open sets===

'''Properties of neighborhoods and open sets'''

Every TVS is [[Connected space|connected]]{{sfn|Narici|Beckenstein|2011|pp=67-113}} and [[Locally connected space|locally connected]]{{sfn|Schaefer|Wolff|1999|p=35}} and any connected open subset of a TVS is [[arcwise connected]]. If <math>S \subseteq X</math> and <math>U</math> is an open subset of <math>X</math> then <math>S + U</math> is an open set in <math>X</math>{{sfn|Narici|Beckenstein|2011|pp=67-113}} and if <math>S \subseteq X</math> has non-empty interior then <math>S - S</math> is a neighborhood of the origin.{{sfn|Narici|Beckenstein|2011|pp=67-113}}

The open convex subsets of a TVS <math>X</math> (not necessarily Hausdorff or locally convex) are exactly those that are of the form <math display=block>z + \{x \in X : p(x) < 1\} ~=~ \{x \in X : p(x - z) < 1\}</math> for some <math>z \in X</math> and some positive continuous [[sublinear functional]] <math>p</math> on <math>X.</math>{{sfn|Narici|Beckenstein|2011|pp=177-220}}

If <math>K</math> is an [[Absorbing set|absorbing]] [[Absolutely convex set|disk]] in a TVS <math>X</math> and if <math>p := p_K</math> is the [[Minkowski functional]] of <math>K</math> then{{sfn|Narici|Beckenstein|2011|p=119-120}} <math display=block>\operatorname{Int}_X K ~\subseteq~ \{x \in X : p(x) < 1\} ~\subseteq~ K ~\subseteq~ \{x \in X : p(x) \leq 1\} ~\subseteq~ \operatorname{cl}_X K</math> where importantly, it was {{em|not}} assumed that <math>K</math> had any topological properties nor that <math>p</math> was continuous (which happens if and only if <math>K</math> is a neighborhood of the origin).

Let <math>\tau</math> and <math>\nu</math> be two vector topologies on <math>X.</math> Then <math>\tau \subseteq \nu</math> if and only if whenever a net <math>x_{\bull} = \left(x_i\right)_{i \in I}</math> in <math>X</math> converges <math>0</math> in <math>(X, \nu)</math> then <math>x_{\bull} \to 0</math> in <math>(X, \tau).</math>{{sfn|Wilansky|2013|p=43}}

Let <math>\mathcal{N}</math> be a neighborhood basis of the origin in <math>X,</math> let <math>S \subseteq X,</math> and let <math>x \in X.</math> Then <math>x \in \operatorname{cl}_X S</math> if and only if there exists a net <math>s_{\bull} = \left(s_N\right)_{N \in \mathcal{N}}</math> in <math>S</math> (indexed by <math>\mathcal{N}</math>) such that <math>s_{\bull} \to x</math> in <math>X.</math>{{sfn|Wilansky|2013|p=42}} This shows, in particular, that it will often suffice to consider nets indexed by a neighborhood basis of the origin rather than nets on arbitrary directed sets.

If <math>X</math> is a TVS that is of the [[second category]] in itself (that is, a [[nonmeager space]]) then any closed convex [[Absorbing set|absorbing]] subset of <math>X</math> is a neighborhood of the origin.{{sfn|Rudin|1991|p=55}} This is no longer guaranteed if the set is not convex (a counter-example exists even in <math>X = \R^2</math>) or if <math>X</math> is not of the second category in itself.{{sfn|Rudin|1991|p=55}}

'''Interior'''

If <math>R, S \subseteq X</math> and <math>S</math> has non-empty interior then 
<math display=block>\operatorname{Int}_X S ~=~ \operatorname{Int}_X \left(\operatorname{cl}_X S\right)~ \text{ and } ~\operatorname{cl}_X S ~=~ \operatorname{cl}_X \left(\operatorname{Int}_X S\right)</math>
and
<math display=block>\operatorname{Int}_X (R) + \operatorname{Int}_X (S) ~\subseteq~ R + \operatorname{Int}_X S \subseteq \operatorname{Int}_X (R + S).</math>

The [[topological interior]] of a [[Absolutely convex set|disk]] is not empty if and only if this interior contains the origin.{{sfn|Narici|Beckenstein|2011|p=108}} 
More generally, if <math>S</math> is a [[Balanced set|balanced]] set with non-empty interior <math>\operatorname{Int}_X S \neq \varnothing</math> in a TVS <math>X</math> then <math>\{0\} \cup \operatorname{Int}_X S</math> will necessarily be balanced;{{sfn|Narici|Beckenstein|2011|pp=67-113}} consequently, <math>\operatorname{Int}_X S</math> will be balanced if and only if it contains the origin.<ref group=proof>This is because every non-empty balanced set must contain the origin and because <math>0 \in \operatorname{Int}_X S</math> if and only if <math>\operatorname{Int}_X S = \{0\} \cup \operatorname{Int}_X S.</math></ref> For this (i.e. <math>0 \in \operatorname{Int}_X S</math>) to be true, it suffices for <math>S</math> to also be convex (in addition to being balanced and having non-empty interior).;{{sfn|Narici|Beckenstein|2011|pp=67-113}} 
The conclusion <math>0 \in \operatorname{Int}_X S</math> could be false if <math>S</math> is not also convex;{{sfn|Narici|Beckenstein|2011|p=108}} for example, in <math>X := \R^2,</math> the interior of the closed and balanced set <math>S := \{(x, y) : x y \geq 0\}</math> is <math>\{(x, y) : x y > 0\}.</math>

If <math>C</math> is convex and <math>0 < t \leq 1,</math> then{{sfn|Jarchow|1981|pp=101-104}} <math>t \operatorname{Int} C + (1 - t) \operatorname{cl} C ~\subseteq~ \operatorname{Int} C.</math> 
Explicitly, this means that if <math>C</math> is a convex subset of a TVS <math>X</math> (not necessarily Hausdorff or locally convex), <math>y \in \operatorname{int}_X C,</math> and <math>x \in \operatorname{cl}_X C</math> then the open line segment joining <math>x</math> and <math>y</math> belongs to the interior of <math>C;</math> that is, <math>\{t x + (1 - t) y : 0 < t < 1\} \subseteq \operatorname{int}_X C.</math>{{sfn|Schaefer|Wolff|1999|p=38}}{{sfn|Conway|1990|p=102}}<ref group=proof>Fix <math>0 < r < 1</math> so it remains to show that <math>w_0 ~\stackrel{\scriptscriptstyle\text{def}}{=}~ r x + (1 - r) y</math> belongs to <math>\operatorname{int}_X C.</math> By replacing <math>C, x, y</math> with <math>C - w_0, x - w_0, y - w_0</math> if necessary, we may assume without loss of generality that <math>r x + (1 - r) y = 0,</math> and so it remains to show that <math>C</math> is a neighborhood of the origin. Let <math>s ~\stackrel{\scriptscriptstyle\text{def}}{=}~ \tfrac{r}{r - 1} < 0</math> so that <math>y = \tfrac{r}{r - 1} x = s x.</math> Since scalar multiplication by <math>s \neq 0</math> is a linear homeomorphism <math>X \to X,</math> <math>\operatorname{cl}_X \left(\tfrac{1}{s} C\right) = \tfrac{1}{s} \operatorname{cl}_X C.</math> Since <math>x \in \operatorname{int} C</math> and <math>y \in \operatorname{cl} C,</math> it follows that <math>x = \tfrac{1}{s} y \in \operatorname{cl} \left(\tfrac{1}{s} C\right) \cap \operatorname{int} C</math> where because <math>\operatorname{int} C</math> is open, there exists some <math>c_0 \in \left(\tfrac{1}{s} C\right) \cap \operatorname{int} C,</math> which satisfies <math>s c_0 \in C.</math> Define <math>h : X \to X</math> by <math>x \mapsto r x + (1 - r) s c_0 = r x - r c_0,</math> which is a homeomorphism because <math>0 < r < 1.</math> The set <math>h\left(\operatorname{int} C\right)</math> is thus an open subset of <math>X</math> that moreover contains <math display=inline>h(c_0) = r c_0 - r c_0 = 0.</math> If <math>c \in \operatorname{int} C</math> then <math display=inline>h(c) = r c + (1 - r) s c_0 \in C</math> since <math>C</math> is convex, <math>0 < r < 1,</math> and <math>s c_0, c \in C,</math> which proves that <math>h\left(\operatorname{int} C\right) \subseteq C.</math> Thus <math>h\left(\operatorname{int} C\right)</math> is an open subset of <math>X</math> that contains the origin and is contained in <math>C.</math> Q.E.D.</ref>

If <math>N \subseteq X</math> is any balanced neighborhood of the origin in <math>X</math> then <math display=inline>\operatorname{Int}_X N \subseteq B_1 N = \bigcup_{0 < |a| < 1} a N \subseteq N</math> where <math>B_1</math> is the set of all scalars <math>a</math> such that <math>|a| < 1.</math>

If <math>x</math> belongs to the interior of a convex set <math>S \subseteq X</math> and <math>y \in \operatorname{cl}_X S,</math> then the half-open line segment <math display=block>[x, y) := \{t x + (1 - t) y : 0 < t \leq 1\} \subseteq \operatorname{Int}_X \text{ if } x \neq y</math> and{{sfn|Schaefer|Wolff|1999|p=38}} <math display=block>[x, x) = \varnothing \text{ if } x = y.</math> 
If <math>N</math> is a [[Balanced set|balanced]] neighborhood of <math>0</math> in <math>X</math> and <math>B_1 := \{a \in \mathbb{K} : |a| < 1\},</math> then by considering intersections of the form <math>N \cap \R x</math> (which are convex [[Symmetric set|symmetric]] neighborhoods of <math>0</math> in the real TVS <math>\R x</math>) it follows that: <math>\operatorname{Int} N = [0, 1) \operatorname{Int} N = (-1, 1) N = B_1 N,</math> and furthermore, if <math>x \in \operatorname{Int} N \text{ and } r := \sup \{r > 0 : [0, r) x \subseteq N\}</math> then <math>r > 1 \text{ and } [0, r) x \subseteq \operatorname{Int} N,</math> and if <math>r \neq \infty</math> then <math>r x \in \operatorname{cl} N \setminus \operatorname{Int} N.</math>

===Non-Hausdorff spaces and the closure of the origin===

A topological vector space <math>X</math> is Hausdorff if and only if <math>\{0\}</math> is a closed subset of <math>X,</math> or equivalently, if and only if <math>\{0\} = \operatorname{cl}_X \{0\}.</math> Because <math>\{0\}</math> is a vector subspace of <math>X,</math> the same is true of its closure <math>\operatorname{cl}_X \{0\},</math> which is referred to as {{em|the closure of the origin}} in <math>X.</math> This vector space satisfies <math display=block>\operatorname{cl}_X \{0\} = \bigcap_{N \in \mathcal{N}(0)} N</math> so that in particular, every neighborhood of the origin in <math>X</math> contains the vector space <math>\operatorname{cl}_X \{0\}</math> as a subset. 
The [[subspace topology]] on <math>\operatorname{cl}_X \{0\}</math> is always the [[trivial topology]], which in particular implies that the topological vector space <math>\operatorname{cl}_X \{0\}</math> a [[compact space]] (even if its dimension is non-zero or even infinite) and consequently also a [[Bounded set (topological vector space)|bounded subset]] of <math>X.</math> In fact, a vector subspace of a TVS is bounded if and only if it is contained in the closure of <math>\{0\}.</math>{{sfn|Narici|Beckenstein|2011|pp=155-176}} 
Every subset of <math>\operatorname{cl}_X \{0\}</math> also carries the trivial topology and so is itself a compact, and thus also complete, [[Topological subspace|subspace]] (see footnote for a proof).<ref group="proof">Since <math>\operatorname{cl}_X \{0\}</math> has the trivial topology, so does each of its subsets, which makes them all compact. It is known that a subset of any uniform space is compact if and only if it is complete and totally bounded.</ref> In particular, if <math>X</math> is not Hausdorff then there exist subsets that are both {{em|compact and complete}} but {{em|not closed}} in <math>X</math>;{{sfn|Narici|Beckenstein|2011|pp=47-66}} for instance, this will be true of any non-empty proper subset of <math>\operatorname{cl}_X \{0\}.</math>

If <math>S \subseteq X</math> is compact, then <math>\operatorname{cl}_X S = S + \operatorname{cl}_X \{0\}</math> and this set is compact. Thus the closure of a compact subset of a TVS is compact (said differently, all compact sets are [[relatively compact]]),{{sfn|Narici|Beckenstein|2011|p=156}} which is not guaranteed for arbitrary non-Hausdorff [[topological space]]s.<ref group="note">In general topology, the closure of a compact subset of a non-Hausdorff space may fail to be compact (for example, the [[particular point topology]] on an infinite set). This result shows that this does not happen in non-Hausdorff TVSs. <math>S + \operatorname{cl}_X \{0\}</math> is compact because it is the image of the compact set <math>S \times \operatorname{cl}_X \{0\}</math> under the continuous addition map <math>\cdot\, + \,\cdot\; : X \times X \to X.</math> Recall also that the sum of a compact set (that is, <math>S</math>) and a closed set is closed so <math>S + \operatorname{cl}_X \{0\}</math> is closed in <math>X.</math></ref>

For every subset <math>S \subseteq X,</math> <math display=block>S + \operatorname{cl}_X \{0\} \subseteq \operatorname{cl}_X S</math> and consequently, if <math>S \subseteq X</math> is open or closed in <math>X</math> then <math>S + \operatorname{cl}_X \{0\} = S</math><ref group="proof" name="ProofSumOfSetAndClosureOf0">If <math>s \in S</math> then <math>s + \operatorname{cl}_X \{0\} = \operatorname{cl}_X (s + \{0\}) = \operatorname{cl}_X \{s\} \subseteq \operatorname{cl}_X S.</math> Because <math>S \subseteq S + \operatorname{cl}_X \{0\} \subseteq \operatorname{cl}_X S,</math> if <math>S</math> is closed then equality holds. Using the fact that <math>\operatorname{cl}_X \{0\}</math> is a vector space, it is readily verified that the complement in <math>X</math> of any set <math>S</math> satisfying the equality <math>S + \operatorname{cl}_X \{0\} = S</math> must also satisfy this equality (when <math>X \setminus S</math> is substituted for <math>S</math>).</ref> (so that this {{em|arbitrary}} open {{em|or}} closed subsets <math>S</math> can be described as a [[Tube lemma|"tube"]] whose vertical side is the vector space <math>\operatorname{cl}_X \{0\}</math>). 
For any subset <math>S \subseteq X</math> of this TVS <math>X,</math> the following are equivalent:

* <math>S</math> is [[Totally bounded space|totally bounded]].
* <math>S + \operatorname{cl}_X \{0\}</math> is totally bounded.{{sfn|Schaefer|Wolff|1999|pp=12-35}}
* <math>\operatorname{cl}_X S</math> is totally bounded.{{sfn|Schaefer|Wolff|1999|p=25}}{{sfn|Jarchow|1981|pp=56-73}}
* The image if <math>S</math> under the canonical quotient map <math>X \to X / \operatorname{cl}_X (\{0\})</math> is totally bounded.{{sfn|Schaefer|Wolff|1999|pp=12-35}}

If <math>M</math> is a vector subspace of a TVS <math>X</math> then <math>X / M</math> is Hausdorff if and only if <math>M</math> is closed in <math>X.</math> 
Moreover, the [[quotient map]] <math>q : X \to X / \operatorname{cl}_X \{0\}</math> is always a [[Open and closed maps|closed map]] onto the (necessarily) Hausdorff TVS.{{sfn|Narici|Beckenstein|2011|pp=107-112}}

Every vector subspace of <math>X</math> that is an algebraic complement of <math>\operatorname{cl}_X \{0\}</math> (that is, a vector subspace <math>H</math> that satisfies <math>\{0\} = H \cap \operatorname{cl}_X \{0\}</math> and <math>X = H + \operatorname{cl}_X \{0\}</math>) is a [[Complemented subspace|topological complement]] of <math>\operatorname{cl}_X \{0\}.</math> 
Consequently, if <math>H</math> is an algebraic complement of <math>\operatorname{cl}_X \{0\}</math> in <math>X</math> then the addition map <math>H \times \operatorname{cl}_X \{0\} \to X,</math> defined by <math>(h, n) \mapsto h + n</math> is a TVS-isomorphism, where <math>H</math> is necessarily Hausdorff and <math>\operatorname{cl}_X \{0\}</math> has the [[indiscrete topology]].{{sfn|Wilansky|2013|p=63}} Moreover, if <math>C</math> is a Hausdorff [[Complete topological vector space|completion]] of <math>H</math> then <math>C \times \operatorname{cl}_X \{0\}</math> is a completion of <math>X \cong H \times \operatorname{cl}_X \{0\}.</math>{{sfn|Schaefer|Wolff|1999|pp=12-35}}

===Closed and compact sets===

'''Compact and totally bounded sets'''

A subset of a TVS is compact if and only if it is complete and [[Totally bounded space|totally bounded]].{{sfn|Narici|Beckenstein|2011|pp=47-66}} Thus, in a [[complete topological vector space]], a closed and totally bounded subset is compact.{{sfn|Narici|Beckenstein|2011|pp=47-66}} 
A subset <math>S</math> of a TVS <math>X</math> is [[Totally bounded space|totally bounded]] if and only if <math>\operatorname{cl}_X S</math> is totally bounded,{{sfn|Schaefer|Wolff|1999|p=25}}{{sfn|Jarchow|1981|pp=56-73}} if and only if its image under the canonical quotient map <math display=block>X \to X / \operatorname{cl}_X (\{0\})</math> is totally bounded.{{sfn|Schaefer|Wolff|1999|pp=12-35}}

Every relatively compact set is totally bounded{{sfn|Narici|Beckenstein|2011|pp=47-66}} and the closure of a totally bounded set is totally bounded.{{sfn|Narici|Beckenstein|2011|pp=47-66}} 
The image of a totally bounded set under a uniformly continuous map (such as a continuous linear map for instance) is totally bounded.{{sfn|Narici|Beckenstein|2011|pp=47-66}} 
If <math>S</math> is a subset of a TVS <math>X</math> such that every sequence in <math>S</math> has a cluster point in <math>S</math> then <math>S</math> is totally bounded.{{sfn|Schaefer|Wolff|1999|pp=12-35}}

If <math>K</math> is a compact subset of a TVS <math>X</math> and <math>U</math> is an open subset of <math>X</math> containing <math>K,</math> then there exists a neighborhood <math>N</math> of 0 such that <math>K + N \subseteq U.</math>{{sfn|Narici|Beckenstein|2011|pp=19-45}}

'''Closure and closed set'''

The closure of any convex (respectively, any balanced, any absorbing) subset of any TVS has this same property. In particular, the closure of any convex, balanced, and absorbing subset is a [[Barrelled space#barrel|barrel]].

The closure of a vector subspace of a TVS is a vector subspace. Every finite dimensional vector subspace of a Hausdorff TVS is closed. 
The sum of a closed vector subspace and a finite-dimensional vector subspace is closed.{{sfn|Narici|Beckenstein|2011|pp=67-113}} 
If <math>M</math> is a vector subspace of <math>X</math> and <math>N</math> is a closed neighborhood of the origin in <math>X</math> such that <math>U \cap N</math> is closed in <math>X</math> then <math>M</math> is closed in <math>X.</math>{{sfn|Narici|Beckenstein|2011|pp=19-45}} 
The sum of a compact set and a closed set is closed. However, the sum of two closed subsets may fail to be closed{{sfn|Narici|Beckenstein|2011|pp=67-113}} (see this footnote<ref group=note>In <math>\R^2,</math> the sum of the <math>y</math>-axis and the graph of <math>y = \frac{1}{x},</math> which is the complement of the <math>y</math>-axis, is open in <math>\R^2.</math> In <math>\R,</math> the [[Minkowski sum]] <math>\Z + \sqrt{2}\Z</math> is a countable dense subset of <math>\R</math> so not closed in <math>\R.</math></ref> for examples). 

If <math>S \subseteq X</math> and <math>a</math> is a scalar then <math display=block>a \operatorname{cl}_X S \subseteq \operatorname{cl}_X (a S),</math> where if <math>X</math> is Hausdorff, <math>a \neq 0, \text{ or } S = \varnothing</math> then equality holds: <math>\operatorname{cl}_X (a S) = a \operatorname{cl}_X S.</math> In particular, every non-zero scalar multiple of a closed set is closed. 
If <math>S \subseteq X</math> and if <math>A</math> is a set of scalars such that neither <math>\operatorname{cl} S \text{ nor } \operatorname{cl} A</math> contain zero then{{sfn|Wilansky|2013|pp=43-44}} <math>\left(\operatorname{cl} A\right) \left(\operatorname{cl}_X S\right) = \operatorname{cl}_X (A S).</math>

If <math>S \subseteq X \text{ and } S + S \subseteq 2 \operatorname{cl}_X S</math> then <math>\operatorname{cl}_X S</math> is convex.{{sfn|Wilansky|2013|pp=43-44}}

If <math>R, S \subseteq X</math> then{{sfn|Narici|Beckenstein|2011|pp=67-113}} <math display=block>\operatorname{cl}_X (R) + \operatorname{cl}_X (S) ~\subseteq~ \operatorname{cl}_X (R + S)~ \text{ and } ~\operatorname{cl}_X \left[ \operatorname{cl}_X (R) + \operatorname{cl}_X (S) \right] ~=~ \operatorname{cl}_X (R + S)</math> and so consequently, if <math>R + S</math> is closed then so is <math>\operatorname{cl}_X (R) + \operatorname{cl}_X (S).</math>{{sfn|Wilansky|2013|pp=43-44}}

If <math>X</math> is a real TVS and <math>S \subseteq X,</math> then <math display=block>\bigcap_{r > 1} r S \subseteq \operatorname{cl}_X S</math> where the left hand side is independent of the topology on <math>X;</math> moreover, if <math>S</math> is a convex neighborhood of the origin then equality holds.

For any subset <math>S \subseteq X,</math> <math display=block>\operatorname{cl}_X S ~=~ \bigcap_{N \in \mathcal{N}} (S + N)</math> where <math>\mathcal{N}</math> is any neighborhood basis at the origin for <math>X.</math>{{sfn|Narici|Beckenstein|2011|pp=80}} 
However, <math display=block>\operatorname{cl}_X U ~\supseteq~ \bigcap \{U : S \subseteq U, U \text{ is open in } X\}</math> and it is possible for this containment to be proper{{sfn|Narici|Beckenstein|2011|pp=108-109}} (for example, if <math>X = \R</math> and <math>S</math> is the rational numbers). It follows that <math>\operatorname{cl}_X U \subseteq U + U</math> for every neighborhood <math>U</math> of the origin in <math>X.</math>{{sfn|Jarchow|1981|pp=30-32}}

'''Closed hulls'''

In a locally convex space, convex hulls of bounded sets are bounded. This is not true for TVSs in general.{{sfn|Narici|Beckenstein|2011|pp=155-176}}

* The closed convex hull of a set is equal to the closure of the convex hull of that set; that is, equal to <math>\operatorname{cl}_X (\operatorname{co} S).</math>{{sfn|Narici|Beckenstein|2011|pp=67-113}}
* The closed balanced hull of a set is equal to the closure of the balanced hull of that set; that is, equal to <math>\operatorname{cl}_X (\operatorname{bal} S).</math>{{sfn|Narici|Beckenstein|2011|pp=67-113}}
* The closed [[Absolutely convex set|disked]] hull of a set is equal to the closure of the disked hull of that set; that is, equal to <math>\operatorname{cl}_X (\operatorname{cobal} S).</math>{{sfn|Narici|Beckenstein|2011|p=109}}

If <math>R, S \subseteq X</math> and the closed convex hull of one of the sets <math>S</math> or <math>R</math> is compact then{{sfn|Narici|Beckenstein|2011|p=109}} <math display=block>\operatorname{cl}_X (\operatorname{co} (R + S)) ~=~  \operatorname{cl}_X (\operatorname{co} R) + \operatorname{cl}_X (\operatorname{co} S).</math> 
If <math>R, S \subseteq X</math> each have a closed convex hull that is compact (that is, <math>\operatorname{cl}_X (\operatorname{co} R)</math> and <math>\operatorname{cl}_X (\operatorname{co} S)</math> are compact) then{{sfn|Narici|Beckenstein|2011|p=109}} <math display=block>\operatorname{cl}_X (\operatorname{co} (R \cup S)) ~=~ \operatorname{co} \left[ \operatorname{cl}_X (\operatorname{co} R) \cup \operatorname{cl}_X (\operatorname{co} S) \right].</math>

'''Hulls and compactness'''

In a general TVS, the closed convex hull of a compact set may {{em|fail}} to be compact. 
The balanced hull of a compact (respectively, [[totally bounded]]) set has that same property.{{sfn|Narici|Beckenstein|2011|pp=67-113}} 
The convex hull of a finite union of compact {{em|convex}} sets is again compact and convex.{{sfn|Narici|Beckenstein|2011|pp=67-113}}

===Other properties===

'''Meager, nowhere dense, and Baire'''

A [[Absolutely convex set|disk]] in a TVS is not [[nowhere dense]] if and only if its closure is a neighborhood of the origin.{{sfn|Narici|Beckenstein|2011|pp=371-423}} 
A vector subspace of a TVS that is closed but not open is [[nowhere dense]].{{sfn|Narici|Beckenstein|2011|pp=371-423}}

Suppose <math>X</math> is a TVS that does not carry the [[indiscrete topology]]. Then <math>X</math> is a [[Baire space]] if and only if <math>X</math> has no balanced absorbing nowhere dense subset.{{sfn|Narici|Beckenstein|2011|pp=371-423}}

A TVS <math>X</math> is a Baire space if and only if <math>X</math> is [[nonmeager]], which happens if and only if there does not exist a [[nowhere dense]] set <math>D</math> such that <math display=inline>X = \bigcup_{n \in \N} n D.</math>{{sfn|Narici|Beckenstein|2011|pp=371-423}} 
Every [[nonmeager]] locally convex TVS is a [[barrelled space]].{{sfn|Narici|Beckenstein|2011|pp=371-423}}

'''Important algebraic facts and common misconceptions'''

If <math>S \subseteq X</math> then <math>2 S \subseteq S + S</math>; if <math>S</math> is convex then equality holds. For an example where equality does {{em|not}} hold, let <math>x</math> be non-zero and set <math>S = \{- x, x\};</math> <math>S = \{x, 2 x\}</math> also works.

A subset <math>C</math> is convex if and only if <math>(s + t) C = s C + t C</math> for all positive real <math>s > 0 \text{ and } t > 0,</math>{{sfn|Rudin|1991|p=38}} or equivalently, if and only if <math>t C + (1 - t) C \subseteq C</math> for all <math>0 \leq t \leq 1.</math>{{sfn|Rudin|1991|p=6}}

The [[convex balanced hull]] of a set <math>S \subseteq X</math> is equal to the convex hull of the [[balanced hull]] of <math>S;</math> that is, it is equal to <math>\operatorname{co} (\operatorname{bal} S).</math> But in general, <math display=block>\operatorname{bal} (\operatorname{co} S) ~\subseteq~ \operatorname{cobal} S ~=~ \operatorname{co} (\operatorname{bal} S),</math> where the inclusion might be strict since the [[balanced hull]] of a convex set need not be convex (counter-examples exist even in <math>\R^2</math>).

If <math>R, S \subseteq X</math> and <math>a</math> is a scalar then{{sfn|Narici|Beckenstein|2011|pp=67-113}} <math display=block>a(R + S) = aR + a S,~ \text{ and } ~\operatorname{co} (R + S) = \operatorname{co} R + \operatorname{co} S,~ \text{ and } ~\operatorname{co} (a S) = a \operatorname{co} S.</math> 
If <math>R, S \subseteq X</math> are convex non-empty disjoint sets and <math>x \not\in R \cup S,</math> then <math>S \cap \operatorname{co} (R \cup \{x\}) = \varnothing </math> or <math>R \cap \operatorname{co} (S \cup \{x\}) = \varnothing.</math>

In any non-trivial vector space <math>X,</math> there exist two disjoint non-empty convex subsets whose union is <math>X.</math>

'''Other properties'''

Every TVS topology can be generated by a {{em|family}} of [[F-seminorm|''F''-seminorms]].{{sfn|Swartz|1992|p=35}} 
<!--START: REMOVED INFO- If <math>f : X \to \R</math> is a subadditive function (that is, <math>f(x + y) \leq f(x) + f(y)</math> for all <math>x, y \in X</math>) such as a [[sublinear function]], [[seminorm]], or [[Linear form|linear functional]], then <math>f</math> is continuous at the origin if and only if it is uniformly continuous on <math>X.</math>{{sfn|Narici|Beckenstein|2011|pp=192-193}} If <math>f : X \to \R</math> is a subadditive and satisfies <math>f(0) = 0</math> then <math>f</math> is continuous if its absolute value <math>|f| : X \to [0, \infty)</math> is continuous. -END:REMOVED INFO-->

If <math>P(x)</math> is some unary [[Predicate (mathematical logic)|predicate]] (a true or false statement dependent on <math>x \in X</math>) then for any <math>z \in X,</math> <math>z + \{x \in X : P(x)\} = \{x \in X : P(x - z)\}.</math><ref group=proof><math display=block>z + \{x \in X : P(x)\} = \{z + x : x \in X, P(x)\} = \{z + x : x \in X, P((z + x) - z)\}</math> and so using <math>y = z + x</math> and the fact that <math>z + X = X,</math> this is equal to <math display=block>\{y : y - z \in X, P(y - z)\} = \{y : y \in X, P(y - z)\} = \{y \in X : P(y - z)\}.</math> [[Q.E.D.]] <math>\blacksquare</math></ref> 
So for example, if <math>P(x)</math> denotes "<math>\|x\| < 1</math>" then for any <math>z \in X,</math> <math>z + \{x \in X : \|x\| < 1\} = \{x \in X : \|x - z\| < 1\}.</math> Similarly, if <math>s \neq 0</math> is a scalar then <math>s \{x \in X : P(x)\} = \left\{x \in X : P\left(\tfrac{1}{s} x\right)\right\}.</math> The elements <math>x \in X</math> of these sets must range over a vector space (that is, over <math>X</math>) rather than not just a subset or else these equalities are no longer guaranteed; similarly, <math>z</math> must belong to this vector space (that is, <math>z \in X</math>).

===Properties preserved by set operators===

* The balanced hull of a compact (respectively, [[totally bounded]], open) set has that same property.{{sfn|Narici|Beckenstein|2011|pp=67-113}}
* The [[Minkowski sum|(Minkowski) sum]] of two compact (respectively, bounded, balanced, convex) sets has that same property.{{sfn|Narici|Beckenstein|2011|pp=67-113}} But the sum of two closed sets need {{em|not}} be closed.
* The convex hull of a balanced (resp. open) set is balanced (respectively, open). However, the convex hull of a closed set need {{em|not}} be closed.{{sfn|Narici|Beckenstein|2011|pp=67-113}} And the convex hull of a bounded set need {{em|not}} be bounded.

The following table, the color of each cell indicates whether or not a given property of subsets of <math>X</math> (indicated by the column name, "convex" for instance) is preserved under the set operator (indicated by the row's name, "closure" for instance). If in every TVS, a property is preserved under the indicated set operator then that cell will be colored green; otherwise, it will be colored red.

So for instance, since the union of two absorbing sets is again absorbing, the cell in row "<math>R \cup S</math>" and column "Absorbing" is colored green. But since the arbitrary intersection of absorbing sets need not be absorbing, the cell in row "Arbitrary intersections (of at least 1 set)" and column "Absorbing" is colored red. If a cell is not colored then that information has yet to be filled in.

{| class="wikitable mw-collapsible mw-collapsed"
|+ Properties preserved by set operators
!rowspan="2"|Operation
!colspan="100"|Property of <math>R,</math> <math>S,</math> and any other subsets of <math>X</math> that is considered
|-
![[Absorbing set|Absorbing]]
![[Balanced set|Balanced]]
![[Convex set|Convex]]
![[Symmetric set|Symmetric]]
!Convex<br />Balanced
!Vector<br />subspace
!Open
!Neighborhood<br />of 0
!Closed
!Closed<br />Balanced
!Closed<br />Convex
!Closed<br />Convex<br />Balanced
![[Barrelled set|Barrel]]
!Closed<br />Vector<br />subspace
![[Totally bounded|Totally<br />bounded]]
![[Compact set|Compact]]
!Compact<br />Convex
![[Relatively compact]]
![[Complete space|Complete]]
![[Sequentially complete space|Sequentially<br />Complete]]
![[Banach disk|Banach<br />disk]]
![[Bounded set (topological vector space)|Bounded]]
![[Bornivorous set|Bornivorous]]
![[Infrabornivorous]]
![[Nowhere dense set|Nowhere<br />dense]] (in <math>X</math>)
![[Meagre set|Meager]]
![[Separable space|Separable]]
![[Metrizable TVS|Pseudometrizable]]
!Operation
|-
!style="text-align:left;"|<math>R \cup S</math>
|{{ya}}<!--Absorbing-->
|{{ya}}<!--Balanced-->
|{{na}}<!--Convex-->
|{{ya}}<!--Symmetric-->
|{{na}}<!--Convex Balanced-->
|{{na}}<!--Vector subspace-->
|{{ya}}<!--Open-->
|{{ya}}<!--Neighborhood of 0-->
|{{ya}}<!--Closed-->
|{{ya}}<!--Closed Balanced-->
|{{na}}<!--Closed Convex-->
|{{na}}<!--Closed Convex Balanced-->
|style="background:;"|<!--Barrel-->
|{{na}}<!--Closed Vector subspace-->
|{{ya}}<!--Totally bounded-->
|{{ya}}<!--Compact-->
|{{na}}<!--Compact convex-->
|{{ya}}<!--Relatively compact-->
|style="background:;"|<!--Complete-->
|{{ya}}<!--Sequentially complete-->
|style="background:;"|<!--Banach disk-->
|{{ya}}<!--Bounded-->
|{{ya}}<!--Bornivorous-->
|{{ya}}<!--Infrabornivorous-->
|{{ya}}<!--Nowhere dense-->
|{{ya}}<!--Meager-->
|{{ya}}<!--Separable-->
|{{ya}}<!--Pseudometrizable-->
!style="text-align:left;"|<math>R \cup S</math>
|-
!style="text-align:left;"|{{nowrap|<math>\cup</math>&nbsp;of}}&nbsp;increasing&nbsp;nonempty&nbsp;[[Chain (order theory)|chain]]
|{{ya}}<!--Absorbing-->
|{{ya}}<!--Balanced-->
|{{ya}}<!--Convex-->
|{{ya}}<!--Symmetric-->
|{{ya}}<!--Convex Balanced-->
|{{ya}}<!--Vector subspace-->
|{{ya}}<!--Open-->
|{{ya}}<!--Neighborhood of 0-->
|{{na}}<!--Closed-->
|{{na}}<!--Closed Balanced-->
|{{na}}<!--Closed Convex-->
|{{na}}<!--Closed Convex Balanced-->
|{{na}}<!--Barrel-->
|{{na}}<!--Closed Vector subspace-->
|{{na}}<!--Totally bounded-->
|{{na}}<!--Compact-->
|{{na}}<!--Compact convex-->
|{{na}}<!--Relatively compact-->
|{{na}}<!--Complete-->
|{{na}}<!--Sequentially complete-->
|style="background:;"|<!--Banach disk-->
|{{na}}<!--Bounded-->
|{{ya}}<!--Bornivorous-->
|{{ya}}<!--Infrabornivorous-->
|{{na}}<!--Nowhere dense-->
|{{na}}<!--Meager-->
|style="background:;"|<!--Separable-->
|style="background:;"|<!--Pseudometrizable-->
!style="text-align:left;"|{{nowrap|<math>\cup</math>&nbsp;of}}&nbsp;increasing&nbsp;nonempty&nbsp;[[Chain (order theory)|chain]]
|-
!style="text-align:left;"|Arbitrary&nbsp;unions (of&nbsp;at&nbsp;least&nbsp;1&nbsp;set)
|{{ya}}<!--Absorbing-->
|{{ya}}<!--Balanced-->
|{{na}}<!--Convex-->
|{{ya}}<!--Symmetric-->
|{{na}}<!--Convex Balanced-->
|{{na}}<!--Vector subspace-->
|{{ya}}<!--Open-->
|{{ya}}<!--Neighborhood of 0-->
|{{na}}<!--Closed-->
|{{na}}<!--Closed Balanced-->
|{{na}}<!--Closed Convex-->
|{{na}}<!--Closed Convex Balanced-->
|{{na}}<!--Barrel-->
|{{na}}<!--Closed Vector subspace-->
|{{na}}<!--Totally bounded-->
|{{na}}<!--Compact-->
|{{na}}<!--Compact convex-->
|{{na}}<!--Relatively compact-->
|{{na}}<!--Complete-->
|{{na}}<!--Sequentially complete-->
|style="background:;"|<!--Banach disk-->
|{{na}}<!--Bounded-->
|{{ya}}<!--Bornivorous-->
|{{ya}}<!--Infrabornivorous-->
|{{na}}<!--Nowhere dense-->
|{{na}}<!--Meager-->
|style="background:;"|<!--Separable-->
|style="background:;"|<!--Pseudometrizable-->
!style="text-align:left;"|Arbitrary&nbsp;unions (of&nbsp;at&nbsp;least&nbsp;1&nbsp;set)
|-
!style="text-align:left;"|<math>R \cap S</math>
|{{ya}}<!--Absorbing-->
|{{ya}}<!--Balanced-->
|{{ya}}<!--Convex-->
|{{ya}}<!--Symmetric-->
|{{ya}}<!--Convex Balanced-->
|{{ya}}<!--Vector subspace-->
|{{ya}}<!--Open-->
|{{ya}}<!--Neighborhood of 0-->
|{{ya}}<!--Closed-->
|{{ya}}<!--Closed Balanced-->
|{{ya}}<!--Closed Convex-->
|{{ya}}<!--Closed Convex Balanced-->
|{{ya}}<!--Barrel-->
|{{ya}}<!--Closed Vector subspace-->
|{{ya}}<!--Totally bounded-->
|{{ya}}<!--Compact-->
|{{ya}}<!--Compact convex-->
|style="background:;"|<!--Relatively compact-->
|style="background:;"|<!--Complete-->
|{{ya}}<!--Sequentially complete-->
|style="background:;"|<!--Banach disk-->
|{{ya}}<!--Bounded-->
|style="background:;"|<!--Bornivorous-->
|style="background:;"|<!--Infrabornivorous-->
|{{ya}}<!--Nowhere dense-->
|{{ya}}<!--Meager-->
|style="background:;"|<!--Separable-->
|{{ya}}<!--Pseudometrizable-->
!style="text-align:left;"|<math>R \cap S</math>
|-
!style="text-align:left;"|{{nowrap|<math>\cap</math>&nbsp;of}}&nbsp;decreasing&nbsp;nonempty&nbsp;[[Chain (order theory)|chain]]
|{{na}}<!--Absorbing-->
|{{ya}}<!--Balanced-->
|{{ya}}<!--Convex-->
|{{ya}}<!--Symmetric-->
|{{ya}}<!--Convex Balanced-->
|{{ya}}<!--Vector subspace-->
|{{na}}<!--Open-->
|{{na}}<!--Neighborhood of 0-->
|{{ya}}<!--Closed-->
|{{ya}}<!--Closed Balanced-->
|{{ya}}<!--Closed Convex-->
|{{ya}}<!--Closed Convex Balanced-->
|{{na}}<!--Barrel-->
|{{ya}}<!--Closed Vector subspace-->
|{{ya}}<!--Totally bounded-->
|style="background:;"|<!--Compact-->
|style="background:;"|<!--Compact convex-->
|style="background:;"|<!--Relatively compact-->
|style="background:;"|<!--Complete-->
|style="background:;"|<!--Sequentially complete-->
|style="background:;"|<!--Banach disk-->
|{{ya}}<!--Bounded-->
|style="background:;"|<!--Bornivorous-->
|style="background:;"|<!--Infrabornivorous-->
|{{ya}}<!--Nowhere dense-->
|style="background:;"|<!--Meager-->
|style="background:;"|<!--Separable-->
|{{ya}}<!--Pseudometrizable-->
!style="text-align:left;"|{{nowrap|<math>\cap</math>&nbsp;of}}&nbsp;decreasing&nbsp;nonempty&nbsp;[[Chain (order theory)|chain]]
|-
!style="text-align:left;"|Arbitrary&nbsp;intersections (of&nbsp;at&nbsp;least&nbsp;1&nbsp;set)
|{{na}}<!--Absorbing-->
|{{ya}}<!--Balanced-->
|{{ya}}<!--Convex-->
|{{ya}}<!--Symmetric-->
|{{ya}}<!--Convex Balanced-->
|{{na}}<!--Open-->
|{{ya}}<!--Vector subspace-->
|{{na}}<!--Neighborhood of 0-->
|{{ya}}<!--Closed-->
|{{ya}}<!--Closed Balanced-->
|{{ya}}<!--Closed Convex-->
|{{ya}}<!--Closed Convex Balanced-->
|{{na}}<!--Barrel-->
|{{ya}}<!--Closed Vector subspace-->
|{{ya}}<!--Totally bounded-->
|style="background:;"|<!--Compact-->
|style="background:;"|<!--Compact convex-->
|style="background:;"|<!--Relatively compact-->
|style="background:;"|<!--Complete-->
|style="background:;"|<!--Sequentially complete-->
|style="background:;"|<!--Banach disk-->
|{{ya}}<!--Bounded-->
|style="background:;"|<!--Bornivorous-->
|style="background:;"|<!--Infrabornivorous-->
|{{ya}}<!--Nowhere dense-->
|style="background:;"|<!--Meager-->
|style="background:;"|<!--Separable-->
|{{ya}}<!--Pseudometrizable-->
!style="text-align:left;"|Arbitrary&nbsp;intersections (of&nbsp;at&nbsp;least&nbsp;1&nbsp;set)
|-
!style="text-align:left;"|<math>R + S</math>
|{{ya}}<!--Absorbing-->
|{{ya}}<!--Balanced-->
|{{ya}}<!--Convex-->
|{{ya}}<!--Symmetric-->
|{{ya}}<!--Convex Balanced-->
|{{ya}}<!--Vector subspace-->
|{{ya}}<!--Open-->
|{{ya}}<!--Neighborhood of 0-->
|{{na}}<!--Closed-->
|style="background:;"|<!--Closed Balanced-->
|{{na}}<!--Closed Convex-->
|style="background:;"|<!--Closed Convex Balanced-->
|style="background:;"|<!--Barrel-->
|style="background:;"|<!--Closed Vector subspace-->
|style="background:;"|<!--Totally bounded-->
|{{ya}}<!--Compact-->
|{{ya}}<!--Compact convex-->
|style="background:;"|<!--Relatively compact-->
|style="background:;"|<!--Complete-->
|style="background:;"|<!--Sequentially complete-->
|style="background:;"|<!--Banach disk-->
|{{ya}}<!--Bounded-->
|style="background:;"|<!--Bornivorous-->
|style="background:;"|<!--Infrabornivorous-->
|style="background:;"|<!--Nowhere dense-->
|style="background:;"|<!--Meager-->
|style="background:;"|<!--Separable-->
|style="background:;"|<!--Pseudometrizable-->
!style="text-align:left;"|<math>R + S</math>
|-
!style="text-align:left;"|Scalar multiple
|{{na}}<!--Absorbing-->
|{{ya}}<!--Balanced-->
|{{ya}}<!--Convex-->
|{{ya}}<!--Symmetric-->
|{{ya}}<!--Convex Balanced-->
|{{ya}}<!--Vector subspace-->
|{{na}}<!--Open-->
|{{na}}<!--Neighborhood of 0-->
|{{na}}<!--Closed-->
|{{na}}<!--Closed Balanced-->
|{{na}}<!--Closed Convex-->
|{{na}}<!--Closed Convex Balanced-->
|{{na}}<!--Barrel-->
|{{na}}<!--Closed Vector subspace-->
|{{ya}}<!--Totally bounded-->
|{{ya}}<!--Compact-->
|{{ya}}<!--Compact convex-->
|{{ya}}<!--Relatively compact-->
|{{ya}}<!--Complete-->
|{{ya}}<!--Sequentially complete-->
|style="background:;"|<!--Banach disk-->
|{{ya}}<!--Bounded-->
|{{na}}<!--Bornivorous-->
|{{na}}<!--Infrabornivorous-->
|{{ya}}<!--Nowhere dense-->
|{{ya}}<!--Meager-->
|{{ya}}<!--Separable-->
|{{ya}}<!--Pseudometrizable-->
!style="text-align:left;"|Scalar multiple
|-
!style="text-align:left;"|Non-0 scalar multiple
|{{ya}}<!--Absorbing-->
|{{ya}}<!--Balanced-->
|{{ya}}<!--Convex-->
|{{ya}}<!--Symmetric-->
|{{ya}}<!--Convex Balanced-->
|{{ya}}<!--Vector subspace-->
|{{ya}}<!--Open-->
|{{ya}}<!--Neighborhood of 0-->
|{{ya}}<!--Closed-->
|{{ya}}<!--Closed Balanced-->
|{{ya}}<!--Closed Convex-->
|{{ya}}<!--Closed Convex Balanced-->
|{{ya}}<!--Barrel-->
|{{ya}}<!--Closed Vector subspace-->
|{{ya}}<!--Totally bounded-->
|{{ya}}<!--Compact-->
|{{ya}}<!--Compact convex-->
|{{ya}}<!--Relatively compact-->
|{{ya}}<!--Complete-->
|{{ya}}<!--Sequentially complete-->
|{{ya}}<!--Banach disk-->
|{{ya}}<!--Bounded-->
|{{ya}}<!--Bornivorous-->
|{{ya}}<!--Infrabornivorous-->
|{{ya}}<!--Nowhere dense-->
|{{ya}}<!--Meager-->
|{{ya}}<!--Separable-->
|{{ya}}<!--Pseudometrizable-->
!style="text-align:left;"|Non-0 scalar multiple
|-
!style="text-align:left;"|Positive scalar multiple
|{{ya}}<!--Absorbing-->
|{{ya}}<!--Balanced-->
|{{ya}}<!--Convex-->
|{{ya}}<!--Symmetric-->
|{{ya}}<!--Convex Balanced-->
|{{ya}}<!--Vector subspace-->
|{{ya}}<!--Open-->
|{{ya}}<!--Neighborhood of 0-->
|{{ya}}<!--Closed-->
|{{ya}}<!--Closed Balanced-->
|{{ya}}<!--Closed Convex-->
|{{ya}}<!--Closed Convex Balanced-->
|{{ya}}<!--Barrel-->
|{{ya}}<!--Closed Vector subspace-->
|{{ya}}<!--Totally bounded-->
|{{ya}}<!--Compact-->
|{{ya}}<!--Compact convex-->
|style="background:;"|<!--Relatively compact-->
|{{ya}}<!--Complete-->
|{{ya}}<!--Sequentially complete-->
|style="background:;"|<!--Banach disk-->
|{{ya}}<!--Bounded-->
|{{ya}}<!--Bornivorous-->
|{{ya}}<!--Infrabornivorous-->
|{{ya}}<!--Nowhere dense-->
|{{ya}}<!--Meager-->
|{{ya}}<!--Separable-->
|{{ya}}<!--Pseudometrizable-->
!style="text-align:left;"|Positive scalar multiple
|-
!style="text-align:left;"|[[Closure (topology)|Closure]]
|{{ya}}<!--Absorbing-->
|{{ya}}<!--Balanced-->
|{{ya}}<!--Convex-->
|{{ya}}<!--Symmetric-->
|{{ya}}<!--Convex Balanced-->
|{{ya}}<!--Vector subspace-->
|{{na}}<!--Open-->
|{{ya}}<!--Neighborhood of 0-->
|{{ya}}<!--Closed-->
|{{ya}}<!--Closed Balanced-->
|{{ya}}<!--Closed Convex-->
|{{ya}}<!--Closed Convex Balanced-->
|{{ya}}<!--Barrel-->
|{{ya}}<!--Closed Vector subspace-->
|{{ya}}<!--Totally bounded-->
|{{ya}}<!--Compact-->
|{{ya}}<!--Compact convex-->
|{{ya}}<!--Relatively compact-->
|style="background:;"|<!--Complete-->
|style="background:;"|<!--Sequentially complete-->
|style="background:;"|<!--Banach disk-->
|{{ya}}<!--Bounded-->
|{{ya}}<!--Bornivorous-->
|{{ya}}<!--Infrabornivorous-->
|{{ya}}<!--Nowhere dense-->
|style="background:;"|<!--Meager-->
|style="background:;"|<!--Separable-->
|style="background:;"|<!--Pseudometrizable-->
!style="text-align:left;"|[[Closure (topology)|Closure]]
|-
!style="text-align:left;"|[[Interior (topology)|Interior]]
|{{na}}<!--Absorbing-->
|{{na}}<!--Balanced-->
|{{ya}}<!--Convex-->
|{{ya}}<!--Symmetric-->
|style="background:;"|<!--Convex Balanced-->
|{{na}}<!--Vector subspace-->
|{{ya}}<!--Open-->
|{{ya}}<!--Neighborhood of 0-->
|{{na}}<!--Closed-->
|{{na}}<!--Closed Balanced-->
|{{na}}<!--Closed Convex-->
|{{na}}<!--Closed Convex Balanced-->
|{{na}}<!--Barrel-->
|{{na}}<!--Closed Vector subspace-->
|style="background:;"|<!--Totally bounded-->
|{{na}}<!--Compact-->
|{{na}}<!--Compact convex-->
|style="background:;"|<!--Relatively compact-->
|style="background:;"|<!--Complete-->
|style="background:;"|<!--Sequentially complete-->
|style="background:;"|<!--Banach disk-->
|{{ya}}<!--Bounded-->
|style="background:;"|<!--Bornivorous-->
|style="background:;"|<!--Infrabornivorous-->
|{{ya}}<!--Nowhere dense-->
|{{na}}<!--Meager-->
|style="background:;"|<!--Separable-->
|style="background:;"|<!--Pseudometrizable-->
!style="text-align:left;"|[[Interior (topology)|Interior]]
|-
!style="text-align:left;"|[[Balanced core]]
|{{ya}}<!--Absorbing-->
|{{ya}}<!--Balanced-->
|{{ya}}<!--Convex-->
|{{ya}}<!--Symmetric-->
|{{ya}}<!--Convex Balanced-->
|{{ya}}<!--Vector subspace-->
|{{ya}}<!--Open-->
|{{ya}}<!--Neighborhood of 0-->
|{{ya}}<!--Closed-->
|{{ya}}<!--Closed Balanced-->
|{{ya}}<!--Closed Convex-->
|{{ya}}<!--Closed Convex Balanced-->
|{{ya}}<!--Barrel-->
|{{ya}}<!--Closed Vector subspace-->
|{{ya}}<!--Totally bounded-->
|style="background:;"|<!--Compact-->
|style="background:;"|<!--Compact convex-->
|style="background:;"|<!--Relatively compact-->
|style="background:;"|<!--Complete-->
|style="background:;"|<!--Sequentially complete-->
|style="background:;"|<!--Banach disk-->
|{{ya}}<!--Bounded-->
|style="background:;"|<!--Bornivorous-->
|style="background:;"|<!--Infrabornivorous-->
|{{ya}}<!--Nowhere dense-->
|{{ya}}<!--Meager-->
|style="background:;"|<!--Separable-->
|style="background:;"|<!--Pseudometrizable-->
!style="text-align:left;"|[[Balanced core]]
|-
!style="text-align:left;"|[[Balanced hull]]
|{{ya}}<!--Absorbing-->
|{{ya}}<!--Balanced-->
|{{na}}<!--Convex-->
|{{ya}}<!--Symmetric-->
|{{ya}}<!--Convex Balanced-->
|{{ya}}<!--Vector subspace-->
|{{ya}}<!--Open-->
|{{ya}}<!--Neighborhood of 0-->
|{{na}}<!--Closed-->
|{{ya}}<!--Closed Balanced-->
|style="background:;"|<!--Closed Convex-->
|{{ya}}<!--Closed Convex Balanced-->
|{{ya}}<!--Barrel-->
|{{ya}}<!--Closed Vector subspace-->
|{{ya}}<!--Totally bounded-->
|{{ya}}<!--Compact-->
|{{na}}<!--Compact convex-->
|style="background:;"|<!--Relatively compact-->
|style="background:;"|<!--Complete-->
|style="background:;"|<!--Sequentially complete-->
|{{ya}}<!--Banach disk-->
|{{ya}}<!--Bounded-->
|{{ya}}<!--Bornivorous-->
|{{ya}}<!--Infrabornivorous-->
|{{na}}<!--Nowhere dense-->
|{{na}}<!--Meager-->
|style="background:;"|<!--Separable-->
|style="background:;"|<!--Pseudometrizable-->
!style="text-align:left;"|[[Balanced hull]]
|-
!style="text-align:left;"|[[Convex hull]]
|{{ya}}<!--Absorbing-->
|{{ya}}<!--Balanced-->
|{{ya}}<!--Convex-->
|{{ya}}<!--Symmetric-->
|{{ya}}<!--Convex Balanced-->
|{{ya}}<!--Vector subspace-->
|{{ya}}<!--Open-->
|{{ya}}<!--Neighborhood of 0-->
|{{na}}<!--Closed-->
|style="background:;"|<!--Closed Balanced-->
|{{ya}}<!--Closed Convex-->
|{{ya}}<!--Closed Convex Balanced-->
|{{ya}}<!--Barrel-->
|{{ya}}<!--Closed Vector subspace-->
|style="background:;"|<!--Totally bounded-->
|style="background:;"|<!--Compact-->
|{{ya}}<!--Compact convex-->
|style="background:;"|<!--Relatively compact-->
|style="background:;"|<!--Complete-->
|style="background:;"|<!--Sequentially complete-->
|{{ya}}<!--Banach disk-->
|{{na}}<!--Bounded-->
|{{ya}}<!--Bornivorous-->
|{{ya}}<!--Infrabornivorous-->
|{{na}}<!--Nowhere dense-->
|{{na}}<!--Meager-->
|style="background:;"|<!--Separable-->
|style="background:;"|<!--Pseudometrizable-->
!style="text-align:left;"|[[Convex hull]]
|-
!style="text-align:left;"|[[Convex balanced hull]]
|{{ya}}<!--Absorbing-->
|{{ya}}<!--Balanced-->
|{{ya}}<!--Convex-->
|{{ya}}<!--Symmetric-->
|{{ya}}<!--Convex Balanced-->
|{{ya}}<!--Vector subspace-->
|{{ya}}<!--Open-->
|{{ya}}<!--Neighborhood of 0-->
|{{ya}}<!--Closed-->
|{{ya}}<!--Closed Balanced-->
|style="background:;"|<!--Closed Convex-->
|{{ya}}<!--Closed Convex Balanced-->
|{{ya}}<!--Barrel-->
|{{ya}}<!--Closed Vector subspace-->
|style="background:;"|<!--Totally bounded-->
|style="background:;"|<!--Compact-->
|style="background:;"|<!--Compact convex-->
|style="background:;"|<!--Relatively compact-->
|style="background:;"|<!--Complete-->
|style="background:;"|<!--Sequentially complete-->
|{{ya}}<!--Banach disk-->
|{{na}}<!--Bounded-->
|{{ya}}<!--Bornivorous-->
|{{ya}}<!--Infrabornivorous-->
|{{na}}<!--Nowhere dense-->
|{{na}}<!--Meager-->
|style="background:;"|<!--Separable-->
|style="background:;"|<!--Pseudometrizable-->
!style="text-align:left;"|[[Convex balanced hull]]
|-
!style="text-align:left;"|Closed balanced hull
|{{ya}}<!--Absorbing-->
|{{ya}}<!--Balanced-->
|{{na}}<!--Convex-->
|{{ya}}<!--Symmetric-->
|{{ya}}<!--Convex Balanced-->
|{{ya}}<!--Vector subspace-->
|{{na}}<!--Open-->
|{{ya}}<!--Neighborhood of 0-->
|{{ya}}<!--Closed-->
|{{ya}}<!--Closed Balanced-->
|{{ya}}<!--Closed Convex-->
|{{ya}}<!--Closed Convex Balanced-->
|{{ya}}<!--Barrel-->
|{{ya}}<!--Closed Vector subspace-->
|style="background:;"|<!--Totally bounded-->
|style="background:;"|<!--Compact-->
|style="background:;"|<!--Compact convex-->
|style="background:;"|<!--Relatively compact-->
|style="background:;"|<!--Complete-->
|style="background:;"|<!--Sequentially complete-->
|style="background:;"|<!--Banach disk-->
|{{ya}}<!--Bounded-->
|{{ya}}<!--Bornivorous-->
|{{ya}}<!--Infrabornivorous-->
|{{na}}<!--Nowhere dense-->
|{{na}}<!--Meager-->
|style="background:;"|<!--Separable-->
|style="background:;"|<!--Pseudometrizable-->
!style="text-align:left;"|Closed balanced hull
|-
!style="text-align:left;"|Closed convex hull
|{{ya}}<!--Absorbing-->
|{{ya}}<!--Balanced-->
|{{ya}}<!--Convex-->
|{{ya}}<!--Symmetric-->
|{{ya}}<!--Convex Balanced-->
|{{ya}}<!--Vector subspace-->
|{{na}}<!--Open-->
|{{ya}}<!--Neighborhood of 0-->
|{{ya}}<!--Closed-->
|{{ya}}<!--Closed Balanced-->
|{{ya}}<!--Closed Convex-->
|{{ya}}<!--Closed Convex Balanced-->
|{{ya}}<!--Barrel-->
|{{ya}}<!--Closed Vector subspace-->
|style="background:;"|<!--Totally bounded-->
|style="background:;"|<!--Compact-->
|style="background:;"|<!--Compact convex-->
|style="background:;"|<!--Relatively compact-->
|style="background:;"|<!--Complete-->
|style="background:;"|<!--Sequentially complete-->
|style="background:;"|<!--Banach disk-->
|{{na}}<!--Bounded-->
|{{ya}}<!--Bornivorous-->
|{{ya}}<!--Infrabornivorous-->
|{{na}}<!--Nowhere dense-->
|{{na}}<!--Meager-->
|style="background:;"|<!--Separable-->
|style="background:;"|<!--Pseudometrizable-->
!style="text-align:left;"|Closed convex hull
|-
!style="text-align:left;"|Closed convex balanced hull
|{{ya}}<!--Absorbing-->
|{{ya}}<!--Balanced-->
|{{ya}}<!--Convex-->
|{{ya}}<!--Symmetric-->
|{{ya}}<!--Convex Balanced-->
|{{ya}}<!--Vector subspace-->
|{{na}}<!--Open-->
|{{ya}}<!--Neighborhood of 0-->
|{{ya}}<!--Closed-->
|{{ya}}<!--Closed Balanced-->
|{{ya}}<!--Closed Convex-->
|{{ya}}<!--Closed Convex Balanced-->
|{{ya}}<!--Barrel-->
|{{ya}}<!--Closed Vector subspace-->
|style="background:;"|<!--Totally bounded-->
|style="background:;"|<!--Compact-->
|style="background:;"|<!--Compact convex-->
|style="background:;"|<!--Relatively compact-->
|style="background:;"|<!--Complete-->
|style="background:;"|<!--Sequentially complete-->
|style="background:;"|<!--Banach disk-->
|{{na}}<!--Bounded-->
|{{ya}}<!--Bornivorous-->
|{{ya}}<!--Infrabornivorous-->
|{{na}}<!--Nowhere dense-->
|{{na}}<!--Meager-->
|style="background:;"|<!--Separable-->
|style="background:;"|<!--Pseudometrizable-->
!style="text-align:left;"|Closed convex balanced hull
|-
!style="text-align:left;"|[[Linear span]]
|{{ya}}<!--Absorbing-->
|{{ya}}<!--Balanced-->
|{{ya}}<!--Convex-->
|{{ya}}<!--Symmetric-->
|{{ya}}<!--Convex Balanced-->
|{{ya}}<!--Vector subspace-->
|{{ya}}<!--Open-->
|{{ya}}<!--Neighborhood of 0-->
|style="background:;"|<!--Closed-->
|style="background:;"|<!--Closed Balanced-->
|style="background:;"|<!--Closed Convex-->
|style="background:;"|<!--Closed Convex Balanced-->
|{{ya}}<!--Barrel-->
|{{ya}}<!--Closed Vector subspace-->
|{{na}}<!--Totally bounded-->
|{{na}}<!--Compact-->
|{{na}}<!--Compact convex-->
|{{na}}<!--Relatively compact-->
|style="background:;"|<!--Complete-->
|style="background:;"|<!--Sequentially complete-->
|{{ya}}<!--Banach disk-->
|{{na}}<!--Bounded-->
|{{ya}}<!--Bornivorous-->
|{{ya}}<!--Infrabornivorous-->
|{{na}}<!--Nowhere dense-->
|{{na}}<!--Meager-->
|style="background:;"|<!--Separable-->
|style="background:;"|<!--Pseudometrizable-->
!style="text-align:left;"|[[Linear span]]
|-
!style="text-align:left;"|Pre-image under a continuous linear map
|{{ya}}<!--Absorbing-->
|{{ya}}<!--Balanced-->
|{{ya}}<!--Convex-->
|{{ya}}<!--Symmetric-->
|{{ya}}<!--Convex Balanced-->
|{{ya}}<!--Vector subspace-->
|{{ya}}<!--Open-->
|{{ya}}<!--Neighborhood of 0-->
|{{ya}}<!--Closed-->
|{{ya}}<!--Closed Balanced-->
|{{ya}}<!--Closed Convex-->
|{{ya}}<!--Closed Convex Balanced-->
|{{ya}}<!--Barrel-->
|{{ya}}<!--Closed Vector subspace-->
|{{na}}<!--Totally bounded-->
|{{na}}<!--Compact-->
|{{na}}<!--Compact convex-->
|{{na}}<!--Relatively compact-->
|style="background:;"|<!--Complete-->
|style="background:;"|<!--Sequentially complete-->
|style="background:;"|<!--Banach disk-->
|{{na}}<!--Bounded-->
|style="background:;"|<!--Bornivorous-->
|style="background:;"|<!--Infrabornivorous-->
|style="background:;"|<!--Nowhere dense-->
|style="background:;"|<!--Meager-->
|{{na}}<!--Separable-->
|{{na}}<!--Pseudometrizable-->
!style="text-align:left;"|Pre-image under a continuous linear map
|-
!style="text-align:left;"|Image under a continuous linear map
|{{na}}<!--Absorbing-->
|{{ya}}<!--Balanced-->
|{{ya}}<!--Convex-->
|{{ya}}<!--Symmetric-->
|{{ya}}<!--Convex Balanced-->
|{{ya}}<!--Vector subspace-->
|{{na}}<!--Open-->
|{{na}}<!--Neighborhood of 0-->
|{{na}}<!--Closed-->
|{{na}}<!--Closed Balanced-->
|{{na}}<!--Closed Convex-->
|{{na}}<!--Closed Convex Balanced-->
|{{na}}<!--Barrel-->
|{{na}}<!--Closed Vector subspace-->
|{{ya}}<!--Totally bounded-->
|{{ya}}<!--Compact-->
|{{ya}}<!--Compact convex-->
|style="background:;"|<!--Relatively compact-->
|style="background:;"|<!--Complete-->
|style="background:;"|<!--Sequentially complete-->
|style="background:;"|<!--Banach disk-->
|{{ya}}<!--Bounded-->
|style="background:;"|<!--Bornivorous-->
|style="background:;"|<!--Infrabornivorous-->
|{{na}}<!--Nowhere dense-->
|style="background:;"|<!--Meager-->
|{{ya}}<!--Separable-->
|style="background:;"|<!--Pseudometrizable-->
!style="text-align:left;"|Image under a continuous linear map
|-
!style="text-align:left;"|Image under a continuous linear surjection
|{{ya}}<!--Absorbing-->
|{{ya}}<!--Balanced-->
|{{ya}}<!--Convex-->
|{{ya}}<!--Symmetric-->
|{{ya}}<!--Convex Balanced-->
|{{ya}}<!--Vector subspace-->
|style="background:;"|<!--Open-->
|style="background:;"|<!--Neighborhood of 0-->
|style="background:;"|<!--Closed-->
|style="background:;"|<!--Closed Balanced-->
|style="background:;"|<!--Closed Convex-->
|style="background:;"|<!--Closed Convex Balanced-->
|style="background:;"|<!--Barrel-->
|style="background:;"|<!--Closed Vector subspace-->
|{{ya}}<!--Totally bounded-->
|{{ya}}<!--Compact-->
|{{ya}}<!--Compact convex-->
|style="background:;"|<!--Relatively compact-->
|style="background:;"|<!--Complete-->
|style="background:;"|<!--Sequentially complete-->
|style="background:;"|<!--Banach disk-->
|{{ya}}<!--Bounded-->
|style="background:;"|<!--Bornivorous-->
|style="background:;"|<!--Infrabornivorous-->
|{{na}}<!--Nowhere dense-->
|style="background:;"|<!--Meager-->
|{{ya}}<!--Separable-->
|style="background:;"|<!--Pseudometrizable-->
!style="text-align:left;"|Image under a continuous linear surjection
|-
!style="text-align:left;"|Non-empty subset of <math>R</math>
|{{na}}<!--Absorbing-->
|{{na}}<!--Balanced-->
|{{na}}<!--Convex-->
|{{na}}<!--Symmetric-->
|{{na}}<!--Convex Balanced-->
|{{na}}<!--Vector subspace-->
|{{na}}<!--Open-->
|{{na}}<!--Neighborhood of 0-->
|{{na}}<!--Closed-->
|{{na}}<!--Closed Balanced-->
|{{na}}<!--Closed Convex-->
|{{na}}<!--Closed Convex Balanced-->
|{{na}}<!--Barrel-->
|{{na}}<!--Closed Vector subspace-->
|{{ya}}<!--Totally bounded-->
|{{na}}<!--Compact-->
|{{na}}<!--Compact convex-->
|style="background:;"|<!--Relatively compact-->
|{{na}}<!--Complete-->
|{{na}}<!--Sequentially complete-->
|style="background:;"|<!--Banach disk-->
|{{ya}}<!--Bounded-->
|{{na}}<!--Bornivorous-->
|{{na}}<!--Infrabornivorous-->
|{{ya}}<!--Nowhere dense-->
|{{ya}}<!--Meager-->
|style="background:;"|<!--Separable-->
|{{ya}}<!--Pseudometrizable-->
!style="text-align:left;"|Non-empty subset of <math>R</math>
|-
!Operation
![[Absorbing set|Absorbing]]
![[Balanced set|Balanced]]
![[Convex set|Convex]]
![[Symmetric set|Symmetric]]
!Convex<br />Balanced
!Vector<br />subspace
!Open
!Neighborhood<br />of 0
!Closed
!Closed<br />Balanced
!Closed<br />Convex
!Closed<br />Convex<br />Balanced
![[Barrelled set|Barrel]]
!Closed<br />Vector<br />subspace
![[Totally bounded|Totally<br />bounded]]
![[Compact set|Compact]]
!Compact<br />Convex
![[Relatively compact]]
![[Complete space|Complete]]
![[Sequentially complete space|Sequentially<br />Complete]]
![[Banach disk|Banach<br />disk]]
![[Bounded set (topological vector space)|Bounded]]
![[Bornivorous set|Bornivorous]]
![[Infrabornivorous]]
![[Nowhere dense set|Nowhere<br />dense]] (in <math>X</math>)
![[Meagre set|Meager]]
![[Separable space|Separable]]
![[Metrizable TVS|Pseudometrizable]]
!Operation
|}

==See also==

* {{annotated link|Banach space}}
* {{annotated link|Complete field}}
* {{annotated link|Hilbert space}}
* {{annotated link|Liquid vector space}}
* {{annotated link|Normed space}}
* {{annotated link|Locally compact field}}
* {{annotated link|Locally compact group}}
* {{annotated link|Locally compact quantum group}}
* {{annotated link|Locally convex topological vector space}}
* {{annotated link|Ordered topological vector space}}
* {{annotated link|Topological abelian group}}
* {{annotated link|Topological field}}
* {{annotated link|Topological group}}
* {{annotated link|Topological module}}
* {{annotated link|Topological ring}}
* {{annotated link|Topological semigroup}}
* {{annotated link|Topological vector lattice}}
* {{annotated link|Measure theory in topological vector spaces}}

==Notes==

{{reflist|group=note}}

===Proofs===

{{reflist|group=proof}}

==Citations==

{{reflist}}

==Bibliography==

{{refbegin|2}}
* {{Adasch Topological Vector Spaces|edition=2}} <!-- {{sfn|Adasch|Ernst|Keim|1978|p=}} -->
* {{Jarchow Locally Convex Spaces}} <!-- {{sfn|Jarchow|1981|p=}} -->
* {{Köthe Topological Vector Spaces I}} <!-- {{sfn|Köthe|1983|p=}} -->
* {{Narici Beckenstein Topological Vector Spaces|edition=2}} <!-- {{sfn|Narici|Beckenstein|2011|p=}} -->
* {{Rudin Walter Functional Analysis|edition=2}} <!-- {{sfn|Rudin|1991|p=}} -->
* {{Schaefer Wolff Topological Vector Spaces|edition=2}} <!-- {{sfn|Schaefer|Wolff|1999|p=}} -->
* {{Schechter Handbook of Analysis and Its Foundations}} <!-- {{sfn|Schechter|1996|p=}} -->
* {{Swartz An Introduction to Functional Analysis}} <!-- {{sfn|Swartz|1992|p=}} -->
* {{Wilansky Modern Methods in Topological Vector Spaces|edition=1}} <!-- {{sfn|Wilansky|2013|p=}} -->
{{refend}}

==Further reading==
* {{Bierstedt An Introduction to Locally Convex Inductive Limits}} <!-- {{sfn|Bierstedt|1988|p=}} -->
* {{Bourbaki Topological Vector Spaces Part 1 Chapters 1–5}} <!-- {{sfn|Bourbaki|1987|p=}} -->
* {{Conway A Course in Functional Analysis|edition=2}} <!-- {{sfn|Conway|1990|p=}} -->
* {{Dunford Schwartz Linear Operators Part 1 General Theory}} <!-- {{sfn|Dunford|1988|p=}} -->
* {{Edwards Functional Analysis Theory and Applications}} <!-- {{sfn|Edwards|1995|p=}} -->
* {{Grothendieck Topological Vector Spaces}} <!-- {{sfn|Grothendieck|1973|p=}} -->
* {{Horváth Topological Vector Spaces and Distributions Volume 1 1966}} <!-- {{sfn|Horváth|1966|p=}} -->
* {{Köthe Topological Vector Spaces II}} <!-- {{sfn|Köthe|1979|p=}} -->
* {{cite book|last=Lang|first=Serge|author-link=Serge Lang|title=Differential manifolds|publisher=Addison-Wesley Publishing Co., Inc.|location=Reading, Mass.&ndash;London&ndash;Don Mills, Ont.|year=1972|isbn=0-201-04166-9}}
* {{Robertson Topological Vector Spaces}} <!-- {{sfn|Robertson|Robertson|1980|p=}} -->
* {{Trèves François Topological vector spaces, distributions and kernels}} <!-- {{sfn|Trèves|2006|p=}} -->
* {{Valdivia Topics in Locally Convex Spaces|edition=1}} <!-- {{sfn|Valdivia|1982|p=}} -->
* {{Voigt A Course on Topological Vector Spaces|edition=1}} <!-- {{sfn|Voigt|2020|p=}} -->

==External links==

* {{Commons category-inline|Topological vector spaces}}

{{Functional Analysis}}
{{TopologicalVectorSpaces}}
{{Authority control}}

[[Category:Articles containing proofs]]
[[Category:Topology of function spaces]]
[[Category:Topological spaces]]
[[Category:Topological vector spaces| ]]
[[Category:Vector spaces]]