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mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It has no generally ...
, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in
functional analysis Image:Drum vibration mode12.gif, 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis. Functional an ...
. A topological vector space is a vector space (an algebraic structure) which is also a
topological space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
, this implies that vector space operations be continuous functions. More specifically, its topological space has a uniform topological structure, allowing a notion of
uniform convergenceIn the mathematical field of analysis, uniform convergence is a mode of Limit of a sequence, convergence of functions stronger than pointwise convergence. A sequence of Function (mathematics), functions (f_n) converges uniformly to a limiting func ...
. The elements of topological vector spaces are typically functions or linear operators acting on topological vector spaces, and the topology is often defined so as to capture a particular notion of convergence of sequences of functions. Banach spaces, Hilbert spaces and Sobolev spaces are well-known examples. Unless stated otherwise, the underlying field of a topological vector space is assumed to be either the
complex number In mathematics, a complex number is a number that can be expressed in the form , where and are real numbers, and is a symbol (mathematics), symbol called the imaginary unit, and satisfying the equation . Because no "real" number satisfies this ...
s \mathbb or the
real number Real may refer to: * Reality, the state of things as they exist, rather than as they may appear or may be thought to be Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish col ...
s \mathbb.


Motivation

;Normed spaces Every
normed vector space The Ateliers et Chantiers de France (ACF, Workshops and Shipyards of France) was a major shipyard that was established in Dunkirk, France, in 1898. The shipyard boomed in the period before World War I (1914–18), but struggled in the inter-war per ...
has a natural topological structure: the norm induces a metric and the metric induces a topology. This is a topological vector space because: #The vector addition is jointly continuous with respect to this topology. This follows directly from the triangle inequality obeyed by the norm. #The scalar multiplication \cdot : \mathbb \times X \rarr X, where \mathbb is the underlying scalar field of , is jointly continuous. This follows from the triangle inequality and homogeneity of the norm. Thus all Banach spaces and Hilbert spaces are examples of topological vector spaces. ;Non-normed spaces There are topological vector spaces whose topology is not induced by a norm, but are still of interest in analysis. Examples of such spaces are spaces of holomorphic functions on an open domain, spaces of infinitely differentiable functions, the Schwartz spaces, and spaces of test functions and the spaces of distribution (mathematics), distributions on them. These are all examples of Montel spaces. An infinite-dimensional Montel space is never normable. The existence of a norm for a given topological vector space is characterized by Kolmogorov's normability criterion. A topological field is a topological vector space over each of its Field extension, subfields.


Definition

Image:Topological vector space illust.svg, A family of neighborhoods of the origin with the above two properties determines uniquely a topological vector space. The system of neighborhoods of any other point in the vector space is obtained by translation (mathematics), translation. A topological vector space (TVS) is a vector space over a topological field \mathbb (most often the real number, real or complex number, complex numbers with their standard topologies) that is endowed with a topological space, topology such that vector addition and scalar multiplication \cdot : \mathbb \times X \rarr X are continuous function (topology), continuous functions (where the domains of these functions are endowed with product topology, product topologies). Such a topology is called a vector topology or a TVS topology on . Every topological vector space is also a commutative topological group under addition. ;Hausdorff assumption Some authors (e.g., Walter Rudin) require the topology on to be T1 space, T1; it then follows that the space is Hausdorff space, Hausdorff, and even Tychonoff space, Tychonoff. A topological vector space is said to be separated if it is Hausdorff; importantly, "separated" does not mean Separable space, separable. The topological and linear algebraic structures can be tied together even more closely with additional assumptions, the most common of which are listed #Types, below. ;Category and morphisms The category (category theory), category of topological vector spaces over a given topological field \mathbb is commonly denoted TVS\mathbb or TVect\mathbb. The object (category theory), objects are the topological vector spaces over \mathbb and the morphisms are the continuous linear map, continuous \mathbb-linear maps from one object to another. A TVS homomorphism or topological homomorphism is a continuous map, continuous linear map between topological vector spaces (TVSs) such that the induced map is an open mapping when , which is the range or image of , is given the subspace topology induced by ''Y''. A TVS embedding or a topological monomorphism is an Injective map, injective topological homomorphism. Equivalently, a TVS-embedding is a linear map that is also a topological embedding. A TVS isomorphism or an isomorphism in the category of TVSs is a bijective Linear map, linear homeomorphism. Equivalently, it is a Surjective map, surjective TVS embedding Many properties of TVSs that are studied, such as Locally convex topological vector space, local convexity, Metrizable TVS, metrizability, Complete topological vector space, completeness, and Normable space, normability, are invariant under TVS isomorphisms. ;A necessary condition for a vector topology A collection of subsets of a vector space is called additive if for every , there exists some such that . All of the above conditions are consequently a necessity for a topology to form a vector topology.


Defining topologies using neighborhoods of the origin

Since every vector topology is translation invariant (i.e. for all , the map defined by is a homeomorphism), to define a vector topology it suffices to define a neighborhood basis (or subbasis) for it at the origin. In general, the set of all balanced and absorbing subsets of a vector space does not satisfy the conditions of this theorem and does not form a neighborhood basis at the origin for any vector topology.


Defining topologies using strings

Let be a vector space and let be a sequence of subsets of . Each set in the sequence is called a knot of and for every index , is called the th knot of . The set is called the beginning of . The sequence is/is a: * Summative if for every index . * Balanced set, Balanced (resp. Absorbing set, absorbing, closed,The topological properties of course also require that be a TVS. convex, open, Symmetric set, symmetric, Barrelled space, barrelled, Absolutely convex set, absolutely convex/disked, etc.) if this is true of every . * String if is summative, absorbing, and balanced. * Topological string or a neighborhood string in a TVS if is a string and each of its knots is a neighborhood of the origin in . If is an Absorbing set, absorbing Absolutely convex set, disk in a vector space then the sequence defined by forms a string beginning with . This is called the natural string of Moreover, if a vector space has countable dimension then every string contains an Absolutely convex set, absolutely convex string. Summative sequences of sets have the particularly nice property that they define non-negative continuous real-valued subadditive functions. These functions can then be used to prove many of the basic properties of topological vector spaces. Assume that always denotes a finite sequence of non-negative integers and use the notation: : and . Observe that for any integers and , :. From this it follows that if consists of distinct positive integers then . It will now be shown by induction on that if consists of non-negative integers such that for some integer then . This is clearly true for and so assume that , which implies that all are positive. If all are distinct then this step is done, otherwise pick distinct indices such that and construct from by replacing with and deleting the element of (all other elements of are transferred to unchanged). Observe that and (since ) so by appealing to the inductive hypothesis, it follows that that , as desired. It is clear that and that so to prove that is subadditive, it suffices to prove that when are such that , which implies that . This is an exercise. If all are symmetric then if and only if from which it follows that and . If all are balanced then the inequality for all unit scalars is proved similarly. Since is a nonnegative subadditive function satisfying , is uniformly continuous on if and only if is continuous at . If all are neighborhoods of the origin then for any real , pick an integer such that so that implies . If all form basis of balanced neighborhoods of the origin then one may show that for any , there exists some such that implies . ∎ If and are two collections of subsets of a vector space and if is a scalar, then by definition: * contains : if and only if for every index . * Set of knots: . * Kernel: . * Scalar multiple: . * Sum: . * Intersection: . If \mathbb is a collection sequences of subsets of , then \mathbb is said to be directed (downwards) under inclusion or simply directed if \mathbb is not empty and for all there exists some such that and (said differently, if and only if \mathbb is a Filter (mathematics), prefilter with respect to the containment defined above). Notation: Let be the set of all knots of all strings in \mathbb. Defining vector topologies using collections of strings is particularly useful for defining classes of TVSs that are not necessarily locally convex. If \mathbb is the set of all topological strings in a TVS then \tau_\mathbb = \tau. A Hausdorff TVS is Metrizable topological vector space, metrizable if and only if its topology can be induced by a single topological string.


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 −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. Let be a topological vector space. Given a Subspace topology, subspace , the quotient space with the usual quotient space (topology), quotient topology is a Hausdorff topological vector space if and only if ''M'' is closed.In particular, is Hausdorff if and only if the set is closed (i.e., is a T1 space, T1 space). This permits the following construction: given a topological vector space (that is probably not Hausdorff), form the quotient space where ''M'' is the closure of . is then a Hausdorff topological vector space that can be studied instead of .


Invariance of vector topologies

One of the most used properties of vector topologies is that every vector topology is translation invariant: :for all , the map defined by is a homeomorphism, but if 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 then the linear map defined by is a homeomorphism. Using produces the negation map defined by , which is consequently a linear homeomorphism and thus a TVS-isomorphism. If and any subset , then and moreover, if then is a Neighborhood (topology), neighborhood (resp. open neighborhood, closed neighborhood) of in if and only if the same is true of at the origin.


Local notions

A subset of a vector space is said to be * Absorbing set, absorbing (in ): if for every , there exists a real such that for any scalar satisfying . * Balanced set, balanced or circled: if for every scalar . * Convex set, convex: if for every real . * a Absolutely convex set, disk or Absolutely convex set, absolutely convex: if is convex and balanced. * Symmetric set, symmetric: if , or equivalently, if . Every neighborhood of 0 is an absorbing set and contains an open Balanced set, balanced neighborhood of so every topological vector space has a local base of absorbing set, absorbing and balanced sets. 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 of a topological vector space is Bounded set (topological vector space), bounded if for every neighborhood , then when is sufficiently large. The definition of boundedness can be weakened a bit; 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. Also, is bounded if and only if for every balanced neighborhood of 0, there exists such that . Moreover, when is locally convex, the boundedness can be characterized by seminorms: the subset is bounded if and only if every continuous seminorm is bounded on . Every totally bounded set is bounded. If is a vector subspace of a TVS , then a subset of is bounded in if and only if it is bounded in .


Metrizability

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 . Let \mathbb 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 \mathbb is locally compact if and only if it is finite-dimensional, that is, isomorphic to \mathbb^n for some natural number .


Completeness and uniform structure

The Complete topological vector space, canonical uniformity on a TVS is the unique translation-invariant Uniform space, uniformity that induces the topology on . Every TVS is assumed to be endowed with this canonical uniformity, which makes all TVSs into uniform spaces. This allows one to about related notions such as Complete topological vector space, completeness,
uniform convergenceIn the mathematical field of analysis, uniform convergence is a mode of Limit of a sequence, convergence of functions stronger than pointwise convergence. A sequence of Function (mathematics), functions (f_n) converges uniformly to a limiting func ...
, 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. 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) is Cauchy if and only if for every neighborhood of , there exists some index such that whenever and . 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. 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. A complete subset of a Hausdorff TVS is closed. * If is a complete subset of a TVS then any subset of that is closed in is complete. * A Cauchy sequence in a Hausdorff TVS is not necessarily relatively compact (i.e. its closure in is not necessarily compact). * If a Cauchy filter in a TVS has an Filters in topology, accumulation point then it converges to . * If a series convergesA series is said to converge in a TVS if the sequence of partial sums converges. in a TVS then in .


Examples


Finest and coarsest vector topology

Let be a real or complex vector space. ;Trivial topology The trivial topology or indiscrete topology is always a TVS topology on any vector space and it is the coarsest TVS topology possible. An important consequence of this is that the intersection of any collection of TVS topologies on 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 . ;Finest vector topology There exists a TVS topology on that is finer than every other TVS-topology on (that is, any TVS-topology on is necessarily a subset of ). Every linear map from into another TVS is necessarily continuous. If has an uncountable Hamel basis then is ''not'' Locally convex topological vector space, locally convex and ''not'' Metrizable topological vector space, metrizable.


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 of all functions f: \mathbb \rarr \mathbb where \mathbb carries its usual Euclidean topology. This set 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 \mathbb^\mathbb,, which carries the natural product topology. With this product topology, becomes a topological vector space whose topology is called . The reason for this name is the following: if is a sequence (or more generally, a Net (mathematics), net) of elements in and if then limit of a sequence, converges to in if and only if for every real number , converges to in \mathbb. 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 with ).


Finite-dimensional spaces

Let \mathbb denote \mathbb or \mathbb and endow \mathbb with its usual Hausdorff normed Euclidean topology. Let be a vector space over \mathbb of finite dimension and so that is vector space isomorphic to \mathbb^n (explicitly, this means that there exists a linear isomorphism between the vector spaces and \mathbb^n). This finite-dimensional vector space always has a unique vector topology, which makes it TVS-isomorphic to \mathbb^n, where \mathbb^n 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 . has a unique vector topology if and only if . If then although does not have a unique vector topology, it does have a unique vector topology. * If then has exactly one vector topology: the trivial topology, which in this case (and 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 . * If then has two vector topologies: the usual Euclidean topology and the (non-Hausdorff) trivial topology. ** Since the field \mathbb is itself a 1-dimensional topological vector space over \mathbb 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
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. The proof of this dichotomy is straightforward so only an outline with the important observations is given. As usual, \mathbb is assumed have the (normed) Euclidean topology. Let be a 1-dimensional vector space over \mathbb. Observe that if B \subseteq \mathbb is a ball centered at 0 and if is a subset containing an "unbounded sequence" then , where an "unbounded sequence" means a sequence of the form where and is unbounded in normed space \mathbb. Any vector topology on will be translation invariant and invariant under non-zero scalar multiplication, and for every , the map given by is a continuous linear bijection. In particular, for any such , X = \mathbb x so every subset of can be written as for some unique subset F \subseteq \mathbb. And if this vector topology on has a neighborhood of 0 that is properly contained in , then the continuity of scalar multiplication \mathbb \times X \rarr X at the origin forces the existence of an open neighborhood of the origin in that ''doesn't'' contain any "unbounded sequence". From this, one deduces that if doesn't carry the trivial topology and if , then for any ball B \subseteq \mathbb center at 0 in \mathbb, contains an open neighborhood of the origin in so that is thus a linear homeomorphism. ∎ * If then has ''infinitely many'' distinct vector topologies: ** Some of these topologies are now described: Every linear functional on , which is vector space isomorphic to \mathbb^n,, induces a seminorm defined by where . Every seminorm induces a (Metrizable TVS, pseudometrizable locally convex) vector topology on and seminorms with distinct kernels induce distinct topologies so that in particular, seminorms on that are induced by linear functionals with distinct kernel will induces distinct vector topologies on . ** However, while there are infinitely many vector topologies on when , there are, ''up to TVS-isomorphism'' only vector topologies on . For instance, if then the vector topologies on consist of the trivial topology, the Hausdorff Euclidean topology, and then the infinitely many remaining non-trivial non-Euclidean vector topologies on are all TVS-isomorphic to one another.


Non-vector topologies

;Discrete and cofinite topologies If is a non-trivial vector space (i.e. of non-zero dimension) then the discrete topology on (which is always Metrizable space, metrizable) is ''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 (where a subset is open if and only if its complement is finite) is also ''not'' a TVS topology on .


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 is continuous if is bounded (as defined below) for some neighborhood of 0. A hyperplane on a topological vector space is either dense or closed. A linear functional on a topological vector space has either dense or closed kernel. Moreover, 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 ''niceness''. * F-spaces are complete space, complete topological vector spaces with a translation-invariant metric. These include Lp space, spaces for all . * Locally convex topological vector spaces: here each point has a local base consisting of convex sets. By a technique known as Minkowski functionals 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 spaces are locally convex (in fact, Banach spaces) for all , but not for . * Barrelled spaces: locally convex spaces where the Banach–Steinhaus theorem holds. * Bornological space: a locally convex space where the continuous linear operators to any locally convex space are exactly the bounded linear operators. * 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 spaces: 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-spaces are limit (category theory), limits of Fréchet spaces. ILH spaces are inverse limits of Hilbert spaces. * Nuclear spaces: 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 spaces and seminormed spaces: 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 spaces: Complete
normed vector space The Ateliers et Chantiers de France (ACF, Workshops and Shipyards of France) was a major shipyard that was established in Dunkirk, France, in 1898. The shipyard boomed in the period before World War I (1914–18), but struggled in the inter-war per ...
s. Most of functional analysis is formulated for Banach spaces. * Reflexive space, Reflexive Banach spaces: 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 ''not'' reflexive is , whose dual is but is strictly contained in the dual of . * Hilbert spaces: 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 spaces. * Euclidean spaces: \mathbb^n or \mathbb^n with the topology induced by the standard inner product. As pointed out in the preceding section, for a given finite , there is only one -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

Every topological vector space has a continuous dual space—the set ''X*'' of all continuous linear functionals, i.e. continuous linear maps from the space into the base field \mathbb. A topology on the dual can be defined to be the coarsest topology such that the dual pairing each point evaluation X* \rarr \mathbb 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 is a not-normable locally convex space, then the pairing map X* \times X \rarr \mathbb is never continuous, no matter which vector space topology one chooses on ''V*''.


Properties

For any of a TVS , the Convex set, convex (resp. Balanced set, balanced, Absolutely convex set, disked, closed convex, closed balanced, closed disked) hull of is the smallest subset of that has this property and contains . The closure (resp. interior, convex hull, balanced hull, disked hull) of a set is sometimes denoted by (resp. , , , ).


Neighborhoods and open sets

;Properties of neighborhoods and open sets * The open convex subsets of a TVS (not necessarily Hausdorff or locally convex) are exactly those that are of the form for some and some positive continuous sublinear functional on . * If and is an open subset of then is an open set in . * If has non-empty interior then is a neighborhood of 0. * If is an Absorbing set, absorbing Absolutely convex set, disk in a TVS and if is the Minkowski functional of then *: ** It was ''not'' assumed that had any topological properties nor that was continuous (which happens if and only if is a neighborhood of 0). * Every TVS is connected space, connected and Locally connected space, locally connected. Any connected open subset of a TVS is arcwise connected. * Let and be two vector topologies on . Then if and only if whenever a net in converges 0 in then in . * Let be a neighborhood basis of the origin in , let , and let . Then if and only if there exists a net in (indexed by ) such that in .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. ;Interior * If has non-empty interior then and . * If and has non-empty interior then . * If is a Absolutely convex set, disk in that has non-empty interior then 0 belongs to the interior of . ** However, a closed Balanced set, balanced subset of with non-empty interior may fail to contain 0 in its interior. * If is a Balanced set, balanced subset of with non-empty interior then is balanced; in particular, if the interior of a balanced set contains the origin then is balanced.If the interior of a balanced set is non-empty but does not contain the origin (such sets exists even in \mathbb^2 and \mathbb^2) then the interior of this set can not be a balanced set. * If belongs to the interior of a convex set and , then the half-open line segment . If is a Balanced set, balanced neighborhood of in then by considering intersections of the form (which are convex Symmetric set, symmetric neighborhoods of in the real TVS ) it follows that: ** , where . ** if and then , , and if then . * If is convex and , then .


Non-Hausdorff spaces and the closure of the origin

* is Hausdorff if and only if is closed in . * so every neighborhood of the origin contains the closure of . * is a vector subspace of and its subspace topology is the trivial topology (which makes compact). * Every subset of is compact and thus complete (see footnote for a proof).Since 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. In particular, if is not Hausdorff then there exist compact complete subsets that are not closed. * for every subset .If then . Since , if is closed then equality holds. Clearly, the complement of any set satisfying the equality must also satisfy this equality. ** So if is open or closed in then (so is a Tube lemma, "tube" with vertical side ). ** The quotient map is a Open and closed maps, closed map onto a Hausdorff TVS. * A subset of a TVS is Totally bounded space, totally bounded if and only if is totally bounded, if and only if is totally bounded, if and only if its image under the canonical quotient map is totally bounded. * If is compact, then and this set is compact. Thus the closure of a compact set is compactIn 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. is compact because it is the image of the compact set under the continuous addition map . Recall also that the sum of a compact set (i.e. ) and a closed set is closed so is closed in . (i.e. all compact sets are relatively compact). * A vector subspace of a TVS is bounded if and only if it is contained in the closure of . * If is a vector subspace of a TVS then is Hausdorff if and only if is closed in . * Every vector subspace of that is an algebraic complement of is a Complemented subspace, topological complement of . Thus if is an algebraic complement of in then the addition map , defined by is a TVS-isomorphism, where is Hausdorff and has the indiscrete topology. Moreover, if is a Hausdorff Complete topological vector space, completion of then is a completion of .


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. ** Thus, in a complete TVS, a closed and totally bounded subset is compact. * A subset of a TVS is Totally bounded space, totally bounded if and only if is totally bounded, if and only if its image under the canonical quotient map is totally bounded. * Every relatively compact set is totally bounded. The closure of a totally bounded set is totally bounded. * The image of a totally bounded set under a uniformly continuous map (e.g. a continuous linear map) is totally bounded. * If is a compact subset of a TVS and is an open subset of containing , then there exists a neighborhood of 0 such that . * If is a subset of a TVS such that every sequence in has a cluster point in then is totally bounded. ;Closure and closed set * If and is a scalar then ; if is Hausdorff, , or then equality holds: . ** In particular, every non-zero scalar multiple of a closed set is closed. * If and then is convex. * If then and . Thus if is closed then so is . * If and if is a set of scalars such that neither nor contain zero then . * The closure of a vector subspace of a TVS is a vector subspace. * If then where is any neighborhood basis at the origin for . ** However, and it's possible for this containment to be proper (e.g. if X = \mathbb and is the rational numbers). ** It follows that for every neighborhood of the origin in . * If is a real TVS and , then (observe that the left hand side is independent of the topology on ); if 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 (see this footnoteIn the \mathbb^2,, the sum of the -axis and the graph of , which is the complement of the -axis, is open in \mathbb^2. In \mathbb, the sum of \mathbb and \sqrt\mathbb is a countable dense subset of \mathbb so not closed in \mathbb. for examples). * If is a vector subspace of and is a closed neighborhood of 0 in such that is closed in then is closed in . * 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. ;Closed hulls * In a locally convex space, convex hulls of bounded sets are bounded. This is not true for TVSs in general. * The closed convex hull of a set is equal to the closure of the convex hull of that set (i.e. to ). * The closed balanced hull of a set is equal to the closure of the balanced hull of that set (i.e. to ). * The closed Absolutely convex set, disked hull of a set is equal to the closure of the disked hull of that set (i.e. to ). * If and the closed convex hull of one of the sets or is compact then . * If each have a closed convex hull that is compact (''i.e.'' and are compact) then . ;Hulls and compactness * In a general TVS, the closed convex hull of a compact set may ''fail'' to be compact. * The balanced hull of a compact (resp. totally bounded) set has that same property. * The convex hull of a finite union of compact ''convex'' sets is again compact and convex.


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. * A vector subspace of a TVS that is closed but not open is nowhere dense. * Suppose is a TVS that does not carry the indiscrete topology. Then is a Baire space if and only if has no balanced absorbing nowhere dense subset. * A TVS is a Baire space if and only if is nonmeager, which happens if and only if there does not exist a nowhere dense set such that . ** Every nonmeager locally convex TVS is a barrelled space. ;Important algebraic facts and common misconceptions * If then ; if is convex then equality holds. ** For an example where equality does ''not'' hold, let be non-zero and set ; also works. * A subset is convex if and only if for all positive real and . * The disked hull of a set is equal to the convex hull of the balanced hull of (i.e. to ). ** However, in general . * If and is a scalar then and . * If are convex non-empty disjoint sets and , then or . * In any non-trivial vector space , there exist two disjoint non-empty convex subsets whose union is . ;Other properties * Every TVS topology can be generated by a ''family'' of ''F''-seminorms.


Properties preserved by set operators

* The balanced hull of a compact (resp. totally bounded, open) set has that same property. * The Minkowski sum, (Minkowski) sum of two compact (resp. bounded, balanced, convex) sets has that same property. But the sum of two closed sets need ''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 ''not'' be closed. And the convex hull of a bounded set need ''not'' be bounded. The following table, the color of each cell indicates whether or not a given property of subsets of (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 "" 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.


See also

* * * * * *


Notes


Citations


References

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External links

* {{Authority control Topological vector spaces, Topology of function spaces