Editing Topological vector space (section)

==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}}