Editing Topological vector space (section)

==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
|}