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←Membuat halaman berisi 'Dalam matematika, sebuah '''ruang vektor topologis''' (atau disebut '''ruang topologis linear''' dan biasanya disingkat '''TVS''' atau '''t.v.s.''') merupakan sala...'
 
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Mendefinisikan topologi vektor menggunakan kumpulan untai khususnya berguna untuk mendefinisikan kelas ruang vektor topologis yang tidak memerlukan cembung lokal.
{{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 &minus;1).
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 (i.e., <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 '''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>.
* '''[[Balanced set|balanced]]''' or '''circled''': if <math>t E \subseteq E</math> for every scalar <math>|t| \leq 1</math>.
* '''[[Convex set|convex]]''': if <math>t E + (1 - t) E \subseteq E</math> for every real <math>0 \leq t \leq 1</math>.
* 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 0 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 set|absorbing]] and [[balanced set]]s.
The origin even has a neighborhood basis consisting of closed balanced neighborhoods of 0; if the space is [[locally convex]] then it also has a neighborhood basis consisting of closed convex balanced neighborhoods of 0.
 
;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, then <math>E \subseteq t V</math> when <math>t</math> is sufficiently large.
 
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 0, 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>.
 
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 three 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]] for 0 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 <math>0</math>.<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>.
 
=== 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 {{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) <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>i</math> such that <math>x_m - x_n \in V</math> whenever <math>j \geq i</math> and <math>k \geq i</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>\sum_{i=1}^{\infty} x_i</math> converges<ref group=note>A series <math>\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>\operatorname{dim} X = 0</math>.
 
;Finest vector topology
 
There exists a TVS topology <math>\tau_f</math> 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}}
 
=== Product vector spaces ===
 
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 (i.e. 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 ===
 
Let <math>\mathbb{K}</math> denote <math>\R</math> or <math>\C</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 := \operatorname{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>\operatorname{dim} X = 0</math>. If <math>\operatorname{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>\operatorname{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 space|Hausdorff]]. The trivial topology on a vector space is Hausdorff if and only if the vector space has dimension <math>0</math>.
* If <math>\operatorname{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 1-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]].
{{collapse top|title=Proof outline|left=true}}
The proof of this dichotomy 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>X</math> be a 1-dimensional vector space over <math>\mathbb{K}</math>. Observe that if <math>B \subseteq \mathbb{K}</math> is a ball centered at 0 and if <math>S \subseteq X</math> is a subset containing an "unbounded sequence" then <math>B \cdot S = X</math>, where an "unbounded sequence" means a sequence of the form <math>s_{\bull} = \left(s_i\right)_{i=1}^{\infty}</math> where <math>0 \neq x \in X</math> and <math>\left(s_i\right)_{i=1}^{\infty} \subseteq \mathbb{K}</math> is unbounded in normed space <math>\mathbb{K}</math>. 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(s) := s x</math> is a continuous linear bijection. In particular, for any such <math>x</math>, <math>X = \mathbb{K} x</math> so 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 of 0 that is properly contained in <math>X</math>, then the continuity of scalar multiplication <math>\mathbb{K} \times X \to X</math> at the origin forces the existence of an open neighborhood of the origin in <math>X</math> that {{em|does not}} contain any "unbounded sequence". From this, one deduces that if <math>X</math> doesn't 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> so that <math>M_x</math> is thus a linear [[homeomorphism]]. ∎
{{collapse bottom}}
* If <math>\operatorname{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>\operatorname{ker} f = \operatorname{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 kernel will induces distinct vector topologies on <math>X</math>.
** However, while there are infinitely many vector topologies on <math>X</math> when <math>\operatorname{dim} X \geq 2</math>, there are, {{em|up to TVS-isomorphism}} only <math>1 + \operatorname{dim} X</math> vector topologies on <math>X</math>. For instance, if <math>n := \operatorname{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 (i.e. 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]] on 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 ordered by their {{em|niceness}}.
 
* [[F-space]]s are [[complete space|complete]] topological vector spaces with a translation-invariant metric. 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. 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. 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. A locally convex F-space is a Fréchet space.
* [[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.
* [[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.
* [[Euclidean space]]s: <math>\R^n</math> or <math>\C^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^{\prime}</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^{\prime} \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]].
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 not-normable locally convex space, then the pairing map <math>X^{\prime} \times X \to \mathbb{K}</math> is never continuous, no matter which vector space topology one chooses on <math>X^{\prime}.</math>
 
== 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 (resp. interior, [[convex hull]], balanced hull, disked hull) of a set <math>S</math> is sometimes denoted by <math>\operatorname{cl}_X S</math> (resp. <math>\operatorname{Int}_X S</math>, <math>\operatorname{co} S</math>, <math>\operatorname{bal} S</math>, <math>\operatorname{cobal} S</math>).
 
=== Neighborhoods and open sets ===
 
;Properties of neighborhoods and open sets
 
* 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>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>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}}
* 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}}
* 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>\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>
** 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 0).
* 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}} Any connected open subset of a TVS is [[arcwise connected]].
* 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 0 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}}<ref group=note>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.</ref>
 
;Interior
 
* If <math>S</math> has non-empty interior then <math>\operatorname{Int}_X S =\operatorname{Int}_X \left(\operatorname{cl}_X S\right)</math> and <math>\operatorname{cl}_X S =\operatorname{cl}_X \left(\operatorname{Int}_X S\right)</math>.
* If <math>R, S \subseteq X</math> and <math>S</math> has non-empty interior then <math>\operatorname{Int}_X (R) + \operatorname{Int}_X (S) \subseteq R + \operatorname{Int}_X S \subseteq \operatorname{Int}_X (R + S)</math>.
* If <math>S</math> is a [[Absolutely convex set|disk]] in <math>X</math> that has non-empty interior then 0 belongs to the interior of <math>S</math>.{{sfn|Narici|Beckenstein|2011|p=108}}
** However, a closed [[Balanced set|balanced]] subset of <math>X</math> with non-empty interior may fail to contain 0 in its interior.{{sfn|Narici|Beckenstein|2011|p=108}}
* If <math>S</math> is a [[Balanced set|balanced]] subset of <math>X</math> with non-empty interior then <math>\{ 0 \} \cup \operatorname{Int}_X S</math> is balanced; in particular, if the interior of a balanced set contains the origin then <math>\operatorname{Int}_X S</math> is balanced.{{sfn|Narici|Beckenstein|2011|pp=67-113}}<ref group="note">If the interior of a balanced set is non-empty but does not contain the origin (such sets exists even in <math>\R^2</math> and <math>\C^2</math>) then the interior of this set can not be a balanced set.</ref>
* 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>[x, y) := \left\{ t x + (1 - t) y : 0 < t \leq 1 \right\}</math> if <math>x \neq y</math> and <math>[x, x) = \varnothing</math> if <math>x = y</math>.{{sfn|Schaefer|Wolff|1999|p=38}} If <math>N</math> is a [[Balanced set|balanced]] neighborhood of <math>0</math> in <math>X</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>, where <math>B_1 := \left\{ a \in \mathbb{K} : |a| < 1 \right\}</math>.
** if <math>x \in \operatorname{Int} N</math> and <math>r := \sup_{} \left\{ r > 0 : [0, r) x \subseteq N \right\}</math> then <math>r > 1</math>, <math>[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>.
* If <math>C</math> is convex and <math>0 < t \leq 1</math>, then <math>t \operatorname{Int} C + (1 - t) \operatorname{cl} C \subseteq \operatorname{Int} C</math>.{{sfn|Jarchow|1981|pp=101-104}}
 
=== Non-Hausdorff spaces and the closure of the origin ===
 
* <math>X</math> is Hausdorff if and only if <math>\{ 0 \}</math> is closed in <math>X</math>.
* <math>\operatorname{cl}_X \{ 0 \} = \bigcap_{N \in \mathcal{N}(0)} N</math> so every neighborhood of the origin contains the closure of <math>\{ 0 \}</math>.
* <math>\operatorname{cl}_X \{ 0 \}</math> is a vector subspace of <math>X</math> and its subspace topology is the [[trivial topology]] (which makes <math>\operatorname{cl}_X \{ 0 \}</math> compact).
* Every subset of <math>\operatorname{cl}_X \{ 0 \}</math> is compact and thus complete (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 compact complete subsets that are not closed.{{sfn|Narici|Beckenstein|2011|pp=47-66}}
* <math>S + \operatorname{cl}_X \{ 0 \} \subseteq \operatorname{cl}_X S</math> for every subset <math>S \subseteq X</math>.<ref group=proof>If <math>s \in S</math> then <math>s + \operatorname{cl}_X \{ 0 \} = \operatorname{cl}_X \left(s + \{ 0 \}\right) = \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. Clearly, the complement of any set <math>S</math> satisfying the equality <math>S + \operatorname{cl}_X \{ 0 \} = S</math> must also satisfy this equality.</ref>
** So if <math>S \subseteq X</math> is open or closed in <math>X</math> then <math>S + \operatorname{cl}_X \{ 0 \} = S</math> (so <math>S</math> is a [[Tube lemma|"tube"]] with vertical side <math>\operatorname{cl}_X \{ 0 \}</math>).
** The [[quotient map]] <math>q : X \to X / \operatorname{cl}_X \{ 0 \}</math> is a [[Open and closed maps|closed map]] onto a Hausdorff TVS.{{sfn|Narici|Beckenstein|2011|pp=107-112}}
* A subset <math>S</math> of a TVS <math>X</math> is [[Totally bounded space|totally bounded]] if and only if <math>S + \operatorname{cl}_X \{ 0 \}</math> is totally bounded,{{sfn|Schaefer|Wolff|1999|pp=12-35}} 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>X \to X / \operatorname{cl}_X \left(\{ 0 \}\right)</math> is totally bounded.{{sfn|Schaefer|Wolff|1999|pp=12-35}}
* 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 set is compact<ref group=note>In general topology, the closure of a compact subset of a non-Hausdorff space may fail to be compact (e.g. 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 (i.e. <math>S</math>) and a closed set is closed so <math>S + \operatorname{cl}_X \{ 0 \}</math> is closed in <math>X</math>.</ref> (i.e. all compact sets are [[relatively compact]]).{{sfn|Narici|Beckenstein|2011|p=156}}
* 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}}
* 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>.
* Every vector subspace of <math>X</math> that is an algebraic complement of <math>\operatorname{cl}_X \{ 0 \}</math> is a [[Complemented subspace|topological complement]] of <math>\operatorname{cl}_X \{ 0 \}</math>. Thus 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 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 TVS, 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>X \to X / \operatorname{cl}_X \left(\{ 0 \}\right)</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}} 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 (e.g. a continuous linear map) is totally bounded.{{sfn|Narici|Beckenstein|2011|pp=47-66}}
* 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}}
* 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}}
 
;Closure and closed set
 
* If <math>S \subseteq X</math> and <math>a</math> is a scalar then <math>a \operatorname{cl}_X S \subseteq \operatorname{cl}_X (a S)</math>; if <math>X</math> is Hausdorff, <math>a \neq 0</math>, or <math>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 <math>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 <math>\operatorname{cl}_X (R) + \operatorname{cl}_X (S) \subseteq \operatorname{cl}_X (R + S)</math> and <math>\operatorname{cl}_X \left[ \operatorname{cl}_X (R) + \operatorname{cl}_X (S) \right] = \operatorname{cl}_X (R + S)</math>.{{sfn|Narici|Beckenstein|2011|pp=67-113}} Thus 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>S \subseteq X</math> and if <math>R</math> is a set of scalars such that neither <math>\operatorname{cl}_X S</math> nor <math>\operatorname{cl}_X R</math> contain zero then <math>\left(\operatorname{cl} R\right) \left(\operatorname{cl}_X S\right) = \operatorname{cl}_X (R S)</math>.{{sfn|Wilansky|2013|pp=43-44}}
* The closure of a vector subspace of a TVS is a vector subspace.
* If <math>S \subseteq X</math> then <math>\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>\operatorname{cl}_X U \supseteq \bigcap \left\{ U : S \subseteq U, U \text{ is open in } X \right\}</math> and it's possible for this containment to be proper{{sfn|Narici|Beckenstein|2011|pp=108-109}} (e.g. 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}}
* If <math>X</math> is a real TVS and <math>S \subseteq X</math>, then <math>\bigcap_{r > 1} r S \subseteq \operatorname{cl}_X S</math> (observe that the left hand side is independent of the topology on <math>X</math>); if <math>S</math> is a convex neighborhood of the origin then equality holds.
* 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 the <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 sum of <math>\mathbb{Z}</math> and <math>\sqrt{2}\mathbb{Z}</math> is a countable dense subset of <math>\R</math> so not closed in <math>\R</math>.</ref> for examples).
* 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}}
* 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}}
 
;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|pp=67-113}}
* 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 <math>\operatorname{cl}_X (\operatorname{co} (R + S)) = \operatorname{cl}_X (\operatorname{co} R) + \operatorname{cl}_X (\operatorname{co} S)</math>.{{sfn|Narici|Beckenstein|2011|pp=67-113}}
* 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 <math>\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 (resp. [[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>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</math> and <math>t.</math>{{sfn|Rudin|1991|p=38}}
* The disked 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, equal to <math>\operatorname{co} (\operatorname{bal} S)</math>. However, in general <math>\operatorname{co} (\operatorname{bal} S) \neq \operatorname{bal} (\operatorname{co} S).</math>
* If <math>R, S \subseteq X</math> and <math>a</math> is a scalar then <math>a(R + S) = aR + a S,</math> and <math>\operatorname{co} (R + S) = \operatorname{co} R + \operatorname{co} S,</math> and <math>\operatorname{co} (a S) = a \operatorname{co} S,</math>{{sfn|Narici|Beckenstein|2011|pp=67-113}}
* 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} \left(R \cup \{ x \}\right) = \varnothing</math> or <math>R \cap \operatorname{co} \left(S \cup \{ x \}\right) = \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''-seminorms.{{sfn|Swartz|1992|p=35}}
<!---- START: REMOVED INFO ----- * If <math>f : X \to \R</math> is a subadditive function (i.e. <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 ------>
 
=== Properties preserved by set operators ===
 
* The balanced hull of a compact (resp. [[totally bounded]], open) set has that same property.{{sfn|Narici|Beckenstein|2011|pp=67-113}}
* The [[Minkowski sum|(Minkowski) sum]] of two compact (resp. 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 (resp. 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 e.g. "convex") is preserved under the set operator (indicated by the row's name e.g. "closure").
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.
 
<div class="NavFrame collapsed">
<div class="NavHead">Properties preserved by set operators</div>
<div class="NavContent" style="text-align:left">
{| class="wikitable" style="border: none; background: none;"
!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>
|style="background:green;"|<!--Absorbing-->
|style="background:green;"|<!--Balanced-->
|style="background:red;"|<!--Convex-->
|style="background:green;"|<!--Symmetric-->
|style="background:red;"|<!--Convex Balanced-->
|style="background:red;"|<!--Vector subspace-->
|style="background:green;"|<!--Open-->
|style="background:green;"|<!--Neighborhood of 0-->
|style="background:green;"|<!--Closed-->
|style="background:green;"|<!--Closed Balanced-->
|style="background:red;"|<!--Closed Convex-->
|style="background:red;"|<!--Closed Convex Balanced-->
|style="background:;"|<!--Barrel-->
|style="background:red;"|<!--Closed Vector subspace-->
|style="background:green;"|<!--Totally bounded-->
|style="background:green;"|<!--Compact-->
|style="background:red;"|<!--Compact convex-->
|style="background:green;"|<!--Relatively compact-->
|style="background:;"|<!--Complete-->
|style="background:green;"|<!--Sequentially complete-->
|style="background:;"|<!--Banach disk-->
|style="background:green;"|<!--Bounded-->
|style="background:green;"|<!--Bornivorous-->
|style="background:green;"|<!--Infrabornivorous-->
|style="background:green;"|<!--Nowhere dense-->
|style="background:green;"|<!--Meager-->
|style="background:green;"|<!--Separable-->
|style="background:green;"|<!--Pseudometrizable-->
!style="text-align:left;"|<math>R \cup S</math>
|-
!style="text-align:left;"|<math>\cup</math>&nbsp;of&nbsp;increasing&nbsp;non-∅&nbsp;chain
|style="background:green;"|<!--Absorbing-->
|style="background:green;"|<!--Balanced-->
|style="background:green;"|<!--Convex-->
|style="background:green;"|<!--Symmetric-->
|style="background:green;"|<!--Convex Balanced-->
|style="background:green;"|<!--Vector subspace-->
|style="background:green;"|<!--Open-->
|style="background:green;"|<!--Neighborhood of 0-->
|style="background:red;"|<!--Closed-->
|style="background:red;"|<!--Closed Balanced-->
|style="background:red;"|<!--Closed Convex-->
|style="background:red;"|<!--Closed Convex Balanced-->
|style="background:red;"|<!--Barrel-->
|style="background:red;"|<!--Closed Vector subspace-->
|style="background:red;"|<!--Totally bounded-->
|style="background:red;"|<!--Compact-->
|style="background:red;"|<!--Compact convex-->
|style="background:red;"|<!--Relatively compact-->
|style="background:red;"|<!--Complete-->
|style="background:red;"|<!--Sequentially complete-->
|style="background:;"|<!--Banach disk-->
|style="background:red;"|<!--Bounded-->
|style="background:green;"|<!--Bornivorous-->
|style="background:green;"|<!--Infrabornivorous-->
|style="background:red;"|<!--Nowhere dense-->
|style="background:red;"|<!--Meager-->
|style="background:;"|<!--Separable-->
|style="background:;"|<!--Pseudometrizable-->
!style="text-align:left;"|<math>\cup</math>&nbsp;of&nbsp;increasing&nbsp;non-∅&nbsp;chain
|-
!style="text-align:left;"|Arbitrary&nbsp;unions (of&nbsp;at&nbsp;least&nbsp;1&nbsp;set)
|style="background:green;"|<!--Absorbing-->
|style="background:green;"|<!--Balanced-->
|style="background:red;"|<!--Convex-->
|style="background:green;"|<!--Symmetric-->
|style="background:red;"|<!--Convex Balanced-->
|style="background:red;"|<!--Vector subspace-->
|style="background:green;"|<!--Open-->
|style="background:green;"|<!--Neighborhood of 0-->
|style="background:red;"|<!--Closed-->
|style="background:red;"|<!--Closed Balanced-->
|style="background:red;"|<!--Closed Convex-->
|style="background:red;"|<!--Closed Convex Balanced-->
|style="background:red;"|<!--Barrel-->
|style="background:red;"|<!--Closed Vector subspace-->
|style="background:red;"|<!--Totally bounded-->
|style="background:red;"|<!--Compact-->
|style="background:red;"|<!--Compact convex-->
|style="background:red;"|<!--Relatively compact-->
|style="background:red;"|<!--Complete-->
|style="background:red;"|<!--Sequentially complete-->
|style="background:;"|<!--Banach disk-->
|style="background:red;"|<!--Bounded-->
|style="background:green;"|<!--Bornivorous-->
|style="background:green;"|<!--Infrabornivorous-->
|style="background:red;"|<!--Nowhere dense-->
|style="background:red;"|<!--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>
|style="background:green;"|<!--Absorbing-->
|style="background:green;"|<!--Balanced-->
|style="background:green;"|<!--Convex-->
|style="background:green;"|<!--Symmetric-->
|style="background:green;"|<!--Convex Balanced-->
|style="background:green;"|<!--Vector subspace-->
|style="background:green;"|<!--Open-->
|style="background:green;"|<!--Neighborhood of 0-->
|style="background:green;"|<!--Closed-->
|style="background:green;"|<!--Closed Balanced-->
|style="background:green;"|<!--Closed Convex-->
|style="background:green;"|<!--Closed Convex Balanced-->
|style="background:green;"|<!--Barrel-->
|style="background:green;"|<!--Closed Vector subspace-->
|style="background:green;"|<!--Totally bounded-->
|style="background:green;"|<!--Compact-->
|style="background:green;"|<!--Compact convex-->
|style="background:;"|<!--Relatively compact-->
|style="background:;"|<!--Complete-->
|style="background:green;"|<!--Sequentially complete-->
|style="background:;"|<!--Banach disk-->
|style="background:green;"|<!--Bounded-->
|style="background:;"|<!--Bornivorous-->
|style="background:;"|<!--Infrabornivorous-->
|style="background:green;"|<!--Nowhere dense-->
|style="background:green;"|<!--Meager-->
|style="background:;"|<!--Separable-->
|style="background:green;"|<!--Pseudometrizable-->
!style="text-align:left;"|<math>R \cap S</math>
|-
!style="text-align:left;"|<math>\cap</math>&nbsp;of&nbsp;decreasing&nbsp;non-<math>\varnothing</math>&nbsp;chain
|style="background:red;"|<!--Absorbing-->
|style="background:green;"|<!--Balanced-->
|style="background:green;"|<!--Convex-->
|style="background:green;"|<!--Symmetric-->
|style="background:green;"|<!--Convex Balanced-->
|style="background:green;"|<!--Vector subspace-->
|style="background:red;"|<!--Open-->
|style="background:red;"|<!--Neighborhood of 0-->
|style="background:green;"|<!--Closed-->
|style="background:green;"|<!--Closed Balanced-->
|style="background:green;"|<!--Closed Convex-->
|style="background:green;"|<!--Closed Convex Balanced-->
|style="background:red;"|<!--Barrel-->
|style="background:green;"|<!--Closed Vector subspace-->
|style="background:green;"|<!--Totally bounded-->
|style="background:;"|<!--Compact-->
|style="background:;"|<!--Compact convex-->
|style="background:;"|<!--Relatively compact-->
|style="background:;"|<!--Complete-->
|style="background:;"|<!--Sequentially complete-->
|style="background:;"|<!--Banach disk-->
|style="background:green;"|<!--Bounded-->
|style="background:;"|<!--Bornivorous-->
|style="background:;"|<!--Infrabornivorous-->
|style="background:green;"|<!--Nowhere dense-->
|style="background:;"|<!--Meager-->
|style="background:;"|<!--Separable-->
|style="background:green;"|<!--Pseudometrizable-->
!style="text-align:left;"|<math>\cap</math>&nbsp;of&nbsp;decreasing&nbsp;non-<math>\varnothing</math>&nbsp;chain
|-
!style="text-align:left;"|Arbitrary&nbsp;intersections (of&nbsp;at&nbsp;least&nbsp;1&nbsp;set)
|style="background:red;"|<!--Absorbing-->
|style="background:green;"|<!--Balanced-->
|style="background:green;"|<!--Convex-->
|style="background:green;"|<!--Symmetric-->
|style="background:green;"|<!--Convex Balanced-->
|style="background:red;"|<!--Open-->
|style="background:green;"|<!--Vector subspace-->
|style="background:red;"|<!--Neighborhood of 0-->
|style="background:green;"|<!--Closed-->
|style="background:green;"|<!--Closed Balanced-->
|style="background:green;"|<!--Closed Convex-->
|style="background:green;"|<!--Closed Convex Balanced-->
|style="background:red;"|<!--Barrel-->
|style="background:green;"|<!--Closed Vector subspace-->
|style="background:green;"|<!--Totally bounded-->
|style="background:;"|<!--Compact-->
|style="background:;"|<!--Compact convex-->
|style="background:;"|<!--Relatively compact-->
|style="background:;"|<!--Complete-->
|style="background:;"|<!--Sequentially complete-->
|style="background:;"|<!--Banach disk-->
|style="background:green;"|<!--Bounded-->
|style="background:;"|<!--Bornivorous-->
|style="background:;"|<!--Infrabornivorous-->
|style="background:green;"|<!--Nowhere dense-->
|style="background:;"|<!--Meager-->
|style="background:;"|<!--Separable-->
|style="background:green;"|<!--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>
|style="background:green;"|<!--Absorbing-->
|style="background:green;"|<!--Balanced-->
|style="background:green;"|<!--Convex-->
|style="background:green;"|<!--Symmetric-->
|style="background:green;"|<!--Convex Balanced-->
|style="background:green;"|<!--Vector subspace-->
|style="background:green;"|<!--Open-->
|style="background:green;"|<!--Neighborhood of 0-->
|style="background:red;"|<!--Closed-->
|style="background:;"|<!--Closed Balanced-->
|style="background:red;"|<!--Closed Convex-->
|style="background:;"|<!--Closed Convex Balanced-->
|style="background:;"|<!--Barrel-->
|style="background:;"|<!--Closed Vector subspace-->
|style="background:;"|<!--Totally bounded-->
|style="background:green;"|<!--Compact-->
|style="background:green;"|<!--Compact convex-->
|style="background:;"|<!--Relatively compact-->
|style="background:;"|<!--Complete-->
|style="background:;"|<!--Sequentially complete-->
|style="background:;"|<!--Banach disk-->
|style="background:green;"|<!--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
|style="background:red;"|<!--Absorbing-->
|style="background:green;"|<!--Balanced-->
|style="background:green;"|<!--Convex-->
|style="background:green;"|<!--Symmetric-->
|style="background:green;"|<!--Convex Balanced-->
|style="background:green;"|<!--Vector subspace-->
|style="background:red;"|<!--Open-->
|style="background:red;"|<!--Neighborhood of 0-->
|style="background:red;"|<!--Closed-->
|style="background:red;"|<!--Closed Balanced-->
|style="background:red;"|<!--Closed Convex-->
|style="background:red;"|<!--Closed Convex Balanced-->
|style="background:red;"|<!--Barrel-->
|style="background:red;"|<!--Closed Vector subspace-->
|style="background:green;"|<!--Totally bounded-->
|style="background:green;"|<!--Compact-->
|style="background:green;"|<!--Compact convex-->
|style="background:green;"|<!--Relatively compact-->
|style="background:green;"|<!--Complete-->
|style="background:green;"|<!--Sequentially complete-->
|style="background:;"|<!--Banach disk-->
|style="background:green;"|<!--Bounded-->
|style="background:red;"|<!--Bornivorous-->
|style="background:red;"|<!--Infrabornivorous-->
|style="background:green;"|<!--Nowhere dense-->
|style="background:green;"|<!--Meager-->
|style="background:green;"|<!--Separable-->
|style="background:green;"|<!--Pseudometrizable-->
!style="text-align:left;"|Scalar multiple
|-
!style="text-align:left;"|Non-0 scalar multiple
|style="background:green;"|<!--Absorbing-->
|style="background:green;"|<!--Balanced-->
|style="background:green;"|<!--Convex-->
|style="background:green;"|<!--Symmetric-->
|style="background:green;"|<!--Convex Balanced-->
|style="background:green;"|<!--Vector subspace-->
|style="background:green;"|<!--Open-->
|style="background:green;"|<!--Neighborhood of 0-->
|style="background:green;"|<!--Closed-->
|style="background:green;"|<!--Closed Balanced-->
|style="background:green;"|<!--Closed Convex-->
|style="background:green;"|<!--Closed Convex Balanced-->
|style="background:green;"|<!--Barrel-->
|style="background:green;"|<!--Closed Vector subspace-->
|style="background:green;"|<!--Totally bounded-->
|style="background:green;"|<!--Compact-->
|style="background:green;"|<!--Compact convex-->
|style="background:green;"|<!--Relatively compact-->
|style="background:green;"|<!--Complete-->
|style="background:green;"|<!--Sequentially complete-->
|style="background:green;"|<!--Banach disk-->
|style="background:green;"|<!--Bounded-->
|style="background:green;"|<!--Bornivorous-->
|style="background:green;"|<!--Infrabornivorous-->
|style="background:green;"|<!--Nowhere dense-->
|style="background:green;"|<!--Meager-->
|style="background:green;"|<!--Separable-->
|style="background:green;"|<!--Pseudometrizable-->
!style="text-align:left;"|Non-0 scalar multiple
|-
!style="text-align:left;"|Positive scalar multiple
|style="background:green;"|<!--Absorbing-->
|style="background:green;"|<!--Balanced-->
|style="background:green;"|<!--Convex-->
|style="background:green;"|<!--Symmetric-->
|style="background:green;"|<!--Convex Balanced-->
|style="background:green;"|<!--Vector subspace-->
|style="background:green;"|<!--Open-->
|style="background:green;"|<!--Neighborhood of 0-->
|style="background:green;"|<!--Closed-->
|style="background:green;"|<!--Closed Balanced-->
|style="background:green;"|<!--Closed Convex-->
|style="background:green;"|<!--Closed Convex Balanced-->
|style="background:green;"|<!--Barrel-->
|style="background:green;"|<!--Closed Vector subspace-->
|style="background:green;"|<!--Totally bounded-->
|style="background:green;"|<!--Compact-->
|style="background:green;"|<!--Compact convex-->
|style="background:;"|<!--Relatively compact-->
|style="background:green;"|<!--Complete-->
|style="background:green;"|<!--Sequentially complete-->
|style="background:;"|<!--Banach disk-->
|style="background:green;"|<!--Bounded-->
|style="background:green;"|<!--Bornivorous-->
|style="background:green;"|<!--Infrabornivorous-->
|style="background:green;"|<!--Nowhere dense-->
|style="background:green;"|<!--Meager-->
|style="background:green;"|<!--Separable-->
|style="background:green;"|<!--Pseudometrizable-->
!style="text-align:left;"|Positive scalar multiple
|-
!style="text-align:left;"|[[Closure (topology)|Closure]]
|style="background:green;"|<!--Absorbing-->
|style="background:green;"|<!--Balanced-->
|style="background:green;"|<!--Convex-->
|style="background:green;"|<!--Symmetric-->
|style="background:green;"|<!--Convex Balanced-->
|style="background:green;"|<!--Vector subspace-->
|style="background:red;"|<!--Open-->
|style="background:green;"|<!--Neighborhood of 0-->
|style="background:green;"|<!--Closed-->
|style="background:green;"|<!--Closed Balanced-->
|style="background:green;"|<!--Closed Convex-->
|style="background:green;"|<!--Closed Convex Balanced-->
|style="background:green;"|<!--Barrel-->
|style="background:green;"|<!--Closed Vector subspace-->
|style="background:green;"|<!--Totally bounded-->
|style="background:green;"|<!--Compact-->
|style="background:green;"|<!--Compact convex-->
|style="background:green;"|<!--Relatively compact-->
|style="background:;"|<!--Complete-->
|style="background:;"|<!--Sequentially complete-->
|style="background:;"|<!--Banach disk-->
|style="background:green;"|<!--Bounded-->
|style="background:green;"|<!--Bornivorous-->
|style="background:green;"|<!--Infrabornivorous-->
|style="background:green;"|<!--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]]
|style="background:red;"|<!--Absorbing-->
|style="background:red;"|<!--Balanced-->
|style="background:green;"|<!--Convex-->
|style="background:green;"|<!--Symmetric-->
|style="background:;"|<!--Convex Balanced-->
|style="background:red;"|<!--Vector subspace-->
|style="background:green;"|<!--Open-->
|style="background:green;"|<!--Neighborhood of 0-->
|style="background:red;"|<!--Closed-->
|style="background:red;"|<!--Closed Balanced-->
|style="background:red;"|<!--Closed Convex-->
|style="background:red;"|<!--Closed Convex Balanced-->
|style="background:red;"|<!--Barrel-->
|style="background:red;"|<!--Closed Vector subspace-->
|style="background:;"|<!--Totally bounded-->
|style="background:red;"|<!--Compact-->
|style="background:red;"|<!--Compact convex-->
|style="background:;"|<!--Relatively compact-->
|style="background:;"|<!--Complete-->
|style="background:;"|<!--Sequentially complete-->
|style="background:;"|<!--Banach disk-->
|style="background:green;"|<!--Bounded-->
|style="background:;"|<!--Bornivorous-->
|style="background:;"|<!--Infrabornivorous-->
|style="background:green;"|<!--Nowhere dense-->
|style="background:red;"|<!--Meager-->
|style="background:;"|<!--Separable-->
|style="background:;"|<!--Pseudometrizable-->
!style="text-align:left;"|[[Interior (topology)|Interior]]
|-
!style="text-align:left;"|[[Balanced core]]
|style="background:green;"|<!--Absorbing-->
|style="background:green;"|<!--Balanced-->
|style="background:green;"|<!--Convex-->
|style="background:green;"|<!--Symmetric-->
|style="background:green;"|<!--Convex Balanced-->
|style="background:green;"|<!--Vector subspace-->
|style="background:green;"|<!--Open-->
|style="background:green;"|<!--Neighborhood of 0-->
|style="background:green;"|<!--Closed-->
|style="background:green;"|<!--Closed Balanced-->
|style="background:green;"|<!--Closed Convex-->
|style="background:green;"|<!--Closed Convex Balanced-->
|style="background:green;"|<!--Barrel-->
|style="background:green;"|<!--Closed Vector subspace-->
|style="background:green;"|<!--Totally bounded-->
|style="background:;"|<!--Compact-->
|style="background:;"|<!--Compact convex-->
|style="background:;"|<!--Relatively compact-->
|style="background:;"|<!--Complete-->
|style="background:;"|<!--Sequentially complete-->
|style="background:;"|<!--Banach disk-->
|style="background:green;"|<!--Bounded-->
|style="background:;"|<!--Bornivorous-->
|style="background:;"|<!--Infrabornivorous-->
|style="background:green;"|<!--Nowhere dense-->
|style="background:green;"|<!--Meager-->
|style="background:;"|<!--Separable-->
|style="background:;"|<!--Pseudometrizable-->
!style="text-align:left;"|[[Balanced core]]
|-
!style="text-align:left;"|Balanced hull
|style="background:green;"|<!--Absorbing-->
|style="background:green;"|<!--Balanced-->
|style="background:red;"|<!--Convex-->
|style="background:green;"|<!--Symmetric-->
|style="background:green;"|<!--Convex Balanced-->
|style="background:green;"|<!--Vector subspace-->
|style="background:green;"|<!--Open-->
|style="background:green;"|<!--Neighborhood of 0-->
|style="background:red;"|<!--Closed-->
|style="background:green;"|<!--Closed Balanced-->
|style="background:;"|<!--Closed Convex-->
|style="background:green;"|<!--Closed Convex Balanced-->
|style="background:green;"|<!--Barrel-->
|style="background:green;"|<!--Closed Vector subspace-->
|style="background:green;"|<!--Totally bounded-->
|style="background:green;"|<!--Compact-->
|style="background:red;"|<!--Compact convex-->
|style="background:;"|<!--Relatively compact-->
|style="background:;"|<!--Complete-->
|style="background:;"|<!--Sequentially complete-->
|style="background:green;"|<!--Banach disk-->
|style="background:green;"|<!--Bounded-->
|style="background:green;"|<!--Bornivorous-->
|style="background:green;"|<!--Infrabornivorous-->
|style="background:red;"|<!--Nowhere dense-->
|style="background:red;"|<!--Meager-->
|style="background:;"|<!--Separable-->
|style="background:;"|<!--Pseudometrizable-->
!style="text-align:left;"|Balanced hull
|-
!style="text-align:left;"|Convex hull
|style="background:green;"|<!--Absorbing-->
|style="background:green;"|<!--Balanced-->
|style="background:green;"|<!--Convex-->
|style="background:;"|<!--Symmetric-->
|style="background:green;"|<!--Convex Balanced-->
|style="background:green;"|<!--Vector subspace-->
|style="background:green;"|<!--Open-->
|style="background:green;"|<!--Neighborhood of 0-->
|style="background:red;"|<!--Closed-->
|style="background:;"|<!--Closed Balanced-->
|style="background:green;"|<!--Closed Convex-->
|style="background:green;"|<!--Closed Convex Balanced-->
|style="background:green;"|<!--Barrel-->
|style="background:green;"|<!--Closed Vector subspace-->
|style="background:;"|<!--Totally bounded-->
|style="background:;"|<!--Compact-->
|style="background:green;"|<!--Compact convex-->
|style="background:;"|<!--Relatively compact-->
|style="background:;"|<!--Complete-->
|style="background:;"|<!--Sequentially complete-->
|style="background:green;"|<!--Banach disk-->
|style="background:red;"|<!--Bounded-->
|style="background:green;"|<!--Bornivorous-->
|style="background:green;"|<!--Infrabornivorous-->
|style="background:red;"|<!--Nowhere dense-->
|style="background:red;"|<!--Meager-->
|style="background:;"|<!--Separable-->
|style="background:;"|<!--Pseudometrizable-->
!style="text-align:left;"|Convex hull
|-
!style="text-align:left;"|Convex balanced hull
|style="background:green;"|<!--Absorbing-->
|style="background:green;"|<!--Balanced-->
|style="background:green;"|<!--Convex-->
|style="background:green;"|<!--Symmetric-->
|style="background:green;"|<!--Convex Balanced-->
|style="background:green;"|<!--Vector subspace-->
|style="background:green;"|<!--Open-->
|style="background:green;"|<!--Neighborhood of 0-->
|style="background:green;"|<!--Closed-->
|style="background:green;"|<!--Closed Balanced-->
|style="background:;"|<!--Closed Convex-->
|style="background:green;"|<!--Closed Convex Balanced-->
|style="background:green;"|<!--Barrel-->
|style="background:green;"|<!--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:green;"|<!--Banach disk-->
|style="background:red;"|<!--Bounded-->
|style="background:green;"|<!--Bornivorous-->
|style="background:green;"|<!--Infrabornivorous-->
|style="background:red;"|<!--Nowhere dense-->
|style="background:red;"|<!--Meager-->
|style="background:;"|<!--Separable-->
|style="background:;"|<!--Pseudometrizable-->
!style="text-align:left;"|Convex balanced hull
|-
!style="text-align:left;"|Closed balanced hull
|style="background:green;"|<!--Absorbing-->
|style="background:green;"|<!--Balanced-->
|style="background:red;"|<!--Convex-->
|style="background:green;"|<!--Symmetric-->
|style="background:green;"|<!--Convex Balanced-->
|style="background:green;"|<!--Vector subspace-->
|style="background:red;"|<!--Open-->
|style="background:green;"|<!--Neighborhood of 0-->
|style="background:green;"|<!--Closed-->
|style="background:green;"|<!--Closed Balanced-->
|style="background:green;"|<!--Closed Convex-->
|style="background:green;"|<!--Closed Convex Balanced-->
|style="background:green;"|<!--Barrel-->
|style="background:green;"|<!--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-->
|style="background:green;"|<!--Bounded-->
|style="background:green;"|<!--Bornivorous-->
|style="background:green;"|<!--Infrabornivorous-->
|style="background:red;"|<!--Nowhere dense-->
|style="background:red;"|<!--Meager-->
|style="background:;"|<!--Separable-->
|style="background:;"|<!--Pseudometrizable-->
!style="text-align:left;"|Closed balanced hull
|-
!style="text-align:left;"|Closed convex hull
|style="background:green;"|<!--Absorbing-->
|style="background:green;"|<!--Balanced-->
|style="background:green;"|<!--Convex-->
|style="background:;"|<!--Symmetric-->
|style="background:green;"|<!--Convex Balanced-->
|style="background:green;"|<!--Vector subspace-->
|style="background:red;"|<!--Open-->
|style="background:green;"|<!--Neighborhood of 0-->
|style="background:green;"|<!--Closed-->
|style="background:green;"|<!--Closed Balanced-->
|style="background:green;"|<!--Closed Convex-->
|style="background:green;"|<!--Closed Convex Balanced-->
|style="background:green;"|<!--Barrel-->
|style="background:green;"|<!--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-->
|style="background:red;"|<!--Bounded-->
|style="background:green;"|<!--Bornivorous-->
|style="background:green;"|<!--Infrabornivorous-->
|style="background:red;"|<!--Nowhere dense-->
|style="background:red;"|<!--Meager-->
|style="background:;"|<!--Separable-->
|style="background:;"|<!--Pseudometrizable-->
!style="text-align:left;"|Closed convex hull
|-
!style="text-align:left;"|Closed convex balanced hull
|style="background:green;"|<!--Absorbing-->
|style="background:green;"|<!--Balanced-->
|style="background:green;"|<!--Convex-->
|style="background:green;"|<!--Symmetric-->
|style="background:green;"|<!--Convex Balanced-->
|style="background:green;"|<!--Vector subspace-->
|style="background:red;"|<!--Open-->
|style="background:green;"|<!--Neighborhood of 0-->
|style="background:green;"|<!--Closed-->
|style="background:green;"|<!--Closed Balanced-->
|style="background:green;"|<!--Closed Convex-->
|style="background:green;"|<!--Closed Convex Balanced-->
|style="background:green;"|<!--Barrel-->
|style="background:green;"|<!--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-->
|style="background:red;"|<!--Bounded-->
|style="background:green;"|<!--Bornivorous-->
|style="background:green;"|<!--Infrabornivorous-->
|style="background:red;"|<!--Nowhere dense-->
|style="background:red;"|<!--Meager-->
|style="background:;"|<!--Separable-->
|style="background:;"|<!--Pseudometrizable-->
!style="text-align:left;"|Closed convex balanced hull
|-
!style="text-align:left;"|Linear span
|style="background:green;"|<!--Absorbing-->
|style="background:green;"|<!--Balanced-->
|style="background:green;"|<!--Convex-->
|style="background:green;"|<!--Symmetric-->
|style="background:green;"|<!--Convex Balanced-->
|style="background:green;"|<!--Vector subspace-->
|style="background:green;"|<!--Open-->
|style="background:green;"|<!--Neighborhood of 0-->
|style="background:;"|<!--Closed-->
|style="background:;"|<!--Closed Balanced-->
|style="background:;"|<!--Closed Convex-->
|style="background:;"|<!--Closed Convex Balanced-->
|style="background:green;"|<!--Barrel-->
|style="background:green;"|<!--Closed Vector subspace-->
|style="background:red;"|<!--Totally bounded-->
|style="background:red;"|<!--Compact-->
|style="background:red;"|<!--Compact convex-->
|style="background:red;"|<!--Relatively compact-->
|style="background:;"|<!--Complete-->
|style="background:;"|<!--Sequentially complete-->
|style="background:green;"|<!--Banach disk-->
|style="background:red;"|<!--Bounded-->
|style="background:green;"|<!--Bornivorous-->
|style="background:green;"|<!--Infrabornivorous-->
|style="background:red;"|<!--Nowhere dense-->
|style="background:red;"|<!--Meager-->
|style="background:;"|<!--Separable-->
|style="background:;"|<!--Pseudometrizable-->
!style="text-align:left;"|Linear span
|-
!style="text-align:left;"|Pre-image under a continuous linear map
|style="background:green;"|<!--Absorbing-->
|style="background:green;"|<!--Balanced-->
|style="background:green;"|<!--Convex-->
|style="background:green;"|<!--Symmetric-->
|style="background:green;"|<!--Convex Balanced-->
|style="background:green;"|<!--Vector subspace-->
|style="background:green;"|<!--Open-->
|style="background:green;"|<!--Neighborhood of 0-->
|style="background:green;"|<!--Closed-->
|style="background:green;"|<!--Closed Balanced-->
|style="background:green;"|<!--Closed Convex-->
|style="background:green;"|<!--Closed Convex Balanced-->
|style="background:green;"|<!--Barrel-->
|style="background:green;"|<!--Closed Vector subspace-->
|style="background:red;"|<!--Totally bounded-->
|style="background:red;"|<!--Compact-->
|style="background:red;"|<!--Compact convex-->
|style="background:red;"|<!--Relatively compact-->
|style="background:;"|<!--Complete-->
|style="background:;"|<!--Sequentially complete-->
|style="background:;"|<!--Banach disk-->
|style="background:red;"|<!--Bounded-->
|style="background:;"|<!--Bornivorous-->
|style="background:;"|<!--Infrabornivorous-->
|style="background:;"|<!--Nowhere dense-->
|style="background:;"|<!--Meager-->
|style="background:red;"|<!--Separable-->
|style="background:red;"|<!--Pseudometrizable-->
!style="text-align:left;"|Pre-image under a continuous linear map
|-
!style="text-align:left;"|Image under a continuous linear map
|style="background:red;"|<!--Absorbing-->
|style="background:green;"|<!--Balanced-->
|style="background:green;"|<!--Convex-->
|style="background:green;"|<!--Symmetric-->
|style="background:green;"|<!--Convex Balanced-->
|style="background:green;"|<!--Vector subspace-->
|style="background:red;"|<!--Open-->
|style="background:red;"|<!--Neighborhood of 0-->
|style="background:red;"|<!--Closed-->
|style="background:red;"|<!--Closed Balanced-->
|style="background:red;"|<!--Closed Convex-->
|style="background:red;"|<!--Closed Convex Balanced-->
|style="background:red;"|<!--Barrel-->
|style="background:red;"|<!--Closed Vector subspace-->
|style="background:green;"|<!--Totally bounded-->
|style="background:green;"|<!--Compact-->
|style="background:green;"|<!--Compact convex-->
|style="background:;"|<!--Relatively compact-->
|style="background:;"|<!--Complete-->
|style="background:;"|<!--Sequentially complete-->
|style="background:;"|<!--Banach disk-->
|style="background:green;"|<!--Bounded-->
|style="background:;"|<!--Bornivorous-->
|style="background:;"|<!--Infrabornivorous-->
|style="background:red;"|<!--Nowhere dense-->
|style="background:;"|<!--Meager-->
|style="background:green;"|<!--Separable-->
|style="background:;"|<!--Pseudometrizable-->
!style="text-align:left;"|Image under a continuous linear map
|-
!style="text-align:left;"|Image under a continuous linear surjection
|style="background:green;"|<!--Absorbing-->
|style="background:green;"|<!--Balanced-->
|style="background:green;"|<!--Convex-->
|style="background:green;"|<!--Symmetric-->
|style="background:green;"|<!--Convex Balanced-->
|style="background:green;"|<!--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-->
|style="background:green;"|<!--Totally bounded-->
|style="background:green;"|<!--Compact-->
|style="background:green;"|<!--Compact convex-->
|style="background:;"|<!--Relatively compact-->
|style="background:;"|<!--Complete-->
|style="background:;"|<!--Sequentially complete-->
|style="background:;"|<!--Banach disk-->
|style="background:green;"|<!--Bounded-->
|style="background:;"|<!--Bornivorous-->
|style="background:;"|<!--Infrabornivorous-->
|style="background:red;"|<!--Nowhere dense-->
|style="background:;"|<!--Meager-->
|style="background:green;"|<!--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>
|style="background:red;"|<!--Absorbing-->
|style="background:red;"|<!--Balanced-->
|style="background:red;"|<!--Convex-->
|style="background:red;"|<!--Symmetric-->
|style="background:red;"|<!--Convex Balanced-->
|style="background:red;"|<!--Vector subspace-->
|style="background:red;"|<!--Open-->
|style="background:red;"|<!--Neighborhood of 0-->
|style="background:red;"|<!--Closed-->
|style="background:red;"|<!--Closed Balanced-->
|style="background:red;"|<!--Closed Convex-->
|style="background:red;"|<!--Closed Convex Balanced-->
|style="background:red;"|<!--Barrel-->
|style="background:red;"|<!--Closed Vector subspace-->
|style="background:green;"|<!--Totally bounded-->
|style="background:red;"|<!--Compact-->
|style="background:red;"|<!--Compact convex-->
|style="background:;"|<!--Relatively compact-->
|style="background:red;"|<!--Complete-->
|style="background:red;"|<!--Sequentially complete-->
|style="background:;"|<!--Banach disk-->
|style="background:green;"|<!--Bounded-->
|style="background:red;"|<!--Bornivorous-->
|style="background:red;"|<!--Infrabornivorous-->
|style="background:green;"|<!--Nowhere dense-->
|style="background:green;"|<!--Meager-->
|style="background:;"|<!--Separable-->
|style="background:green;"|<!--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
|}
</div>
</div>
 
== See also ==
 
* {{annotated link|Banach space}}
* {{annotated link|Hilbert space}}
* {{annotated link|Normed space}}
* {{annotated link|Locally convex topological vector space}}
* {{annotated link|Topological group}}
* {{annotated link|Vector space}}
 
== Notes ==
{{reflist|group=note}}
{{reflist|group=proof}}
 
== Citations ==
{{reflist}}
 
== References ==
{{refbegin|2}}
* {{Adasch Topological Vector Spaces|edition=2}} <!-- {{sfn|Adasch|Ernst|Keim|1978|p=}} -->
* {{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=}} -->
* {{Jarchow Locally Convex Spaces}} <!-- {{sfn|Jarchow|1981|p=}} -->
* {{Köthe Topological Vector Spaces I}} <!-- {{sfn|Köthe|1983|p=}} -->
* {{Köthe Topological Vector Spaces II}} <!-- {{sfn|Köthe|1979|p=}} -->
* {{cite book|last=Köthe|first=Gottfried|author-link=Gottfried Köthe|title=Topological vector spaces I|series=Grundlehren der mathematischen Wissenschaften|volume=159|publisher=[[Springer-Verlag]]|location=New York|orig-year=1969|year=1983|isbn=978-3-642-64990-5}}
* {{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}}
* {{Narici Beckenstein Topological Vector Spaces|edition=2}} <!-- {{sfn|Narici|Beckenstein|2011|p=}} -->
* {{cite book|last1=Robertson|first1=A.P.|last2=Robertson|first2=W.J.|title= Topological vector spaces|series=Cambridge Tracts in Mathematics|volume=53|year=1964|publisher=[[Cambridge University Press]] }}
* {{Robertson Topological Vector Spaces}} <!-- {{sfn|Robertson|Robertson|1980|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=}} -->
* {{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=}} -->
* {{Wilansky Modern Methods in Topological Vector Spaces|edition=1}} <!-- {{sfn|Wilansky|2013|p=}} -->
{{refend}}
 
== External links ==
 
* {{Commons category-inline|Topological vector spaces}}
 
{{Functional Analysis}}
{{TopologicalVectorSpaces}}
{{Authority control}}
 
[[Category:Topological vector spaces| ]]
[[Category:Topology of function spaces]]
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