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

Many topological vector spaces are spaces of functionss, 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.

In this article, the scalar field of a topological vector space will be assumed to be either the complex numbers or the real numbers unless clearly stated otherwise.

Motivation

Normed spaces

Every normed vector space has a natural topological structure: the norm induces a metric and the metric induces a topology. This is a topological vector space because:

1. is (jointly) continuous with respect to this topology. This follows directly from the triangle inequality obeyed by the norm.

2. 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 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 subfields.

Definition

A topological vector space (TVS is a vector space over a topological field (most often the real or complex numbers with their standard topologies) that is endowed with a topology are continuous functions (where the domains of these functions are endowed with product topologies). Such a topology is called a or a

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

Hausdorff assumption

Many authors (for example, Walter Rudin to be T1; it then follows that the space is Hausdorff, and even Tychonoff. A topological vector space is said to be if it is Hausdorff; importantly, "separated" does not mean separable. The topological and linear algebraic structures can be tied together even more closely with additional assumptions, the most common of which are listed below.

Category and morphisms

The category The objects and the morphisms are the \mathbb{K}-linear mapss from one object to another.

A (abbreviated ), also called a , is a continuous linear map is an open mapping is given the subspace topology

A (abbreviated ), also called a , is an injective topological homomorphism. Equivalently, a TVS-embedding is a linear map that is also a topological embedding.

A (abbreviated ), also called a or an , is a bijective linear homeomorphism. Equivalently, it is a surjective TVS embedding

Many properties of TVSs that are studied, such as local convexity, metrizability, completeness, and normability, are invariant under TVS isomorphisms.

A necessary condition for a vector topology

of subsets of a vector space is called

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

Defining topologies using neighborhoods of the origin

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

is called a is called the -th knot is called the beginning is/is a:

• Balanced (resp. absorbing, closed, convex, open, symmetric, barrelled, absolutely convex/disked

• is summative, absorbing, and balanced.

• or a

is an absorbing disk This is called the has countable dimension then every string contains an 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.

A proof of the above theorem is given in the article on metrizable topological vector spaces.

is a scalar, then by definition:

• contains

• Set of knots

• Kernel

• Scalar multiple

• Sum

• Intersection

is said to be directed (downwards) under inclusion or simply directed downward is a prefilter defined above).

Notation

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

A Hausdorff TVS is metrizable if and only if its topology can be induced by a single topological string.

Topological structure

A vector space is an abelian group). Hence, every topological vector space is an abelian topological group. Every TVS is completely regular but a TVS need not be normal.

be a topological vector space. Given a subspace with the usual quotient topology is closed.

Invariance of vector topologies

One of the most used properties of vector topologies is that every vector topology is :

1. is a homeomorphism then it is not linear and so not a TVS-isomorphism.

which is consequently a linear homeomorphism and thus a TVS-isomorphism.

is a neighborhood at the origin.

Local notions

is said to be

• absorbing

• balanced or circled

• convex

• a disk or absolutely convex is convex and balanced.

• symmetric

Every neighborhood of the origin is an absorbing set and contains an open balanced so every topological vector space has a local base of absorbing and balanced sets if the space is locally convex then it also has a neighborhood basis consisting of closed convex balanced neighborhoods of the origin.

Bounded subsets

is bounded.

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.

is locally convex, the boundedness can be characterized by seminorms

Every totally bounded set is bounded.

Metrizability

A TVS is 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 F-seminorm. A TVS is metrizable if and only if it is Hausdorff and pseudometrizable.

More strongly: a topological vector space is said to be normable if its topology can be induced by a norm. A topological vector space is normable if and only if it is Hausdorff and has a convex bounded neighborhood of the origin.

be a non-discrete locally compact topological field, for example the real or complex numbers. A Hausdorff is locally compact if and only if it is finite-dimensional

Completeness and uniform structure

The canonical uniformity is the unique translation-invariant uniformity

Every TVS is assumed to be endowed with this canonical uniformity, which makes all TVSs into uniform spaces. Therefore making sense to related notions such as 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. A subset of a TVS is compact if and only if it is complete and totally bounded (for Hausdorff TVSs, a set being totally bounded is equivalent to it being 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 (or sequence is Cauchy

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 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 linear subspace of a complete topological vector space.

• Every TVS has a 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.

• is complete.

• is not necessarily relatively compact is not necessarily compact).

• If a Cauchy filter in a TVS has an accumulation point

• converges

Examples

Finest and coarsest vector topology

be a real or complex vector space.

Trivial topology

The trivial topology or indiscrete topology 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 pseudometrizable seminormable locally convex topological vector space. It is Hausdorff

Finest vector topology

called the ). has an uncountable Hamel basis is locally convex and metrizable.

Cartesian products

A Cartesian product of a family of topological vector spaces, when endowed with the product topology carries its usual Euclidean topology 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 which carries the natural product topology becomes a topological vector space whose topology is called is a sequence (or more generally, a net converges This TVS is complete, Hausdorff, and locally convex but not metrizable and consequently not normable).

Finite-dimensional spaces

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

with its usual Hausdorff normed Euclidean topology (explicitly, this means that there exists a linear isomorphism always has a unique is endowed with the usual Euclidean topology (which is the same as the product topology). This Hausdorff vector topology is also the (unique) finest does not have a unique vector topology, it does have a unique vector topology.

• has exactly one vector topology: the trivial topology, which in this case (and

has two vector topologies: the usual Euclidean topology and the (non-Hausdorff) trivial topology.

• 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.

has distinct vector topologies:

• induces a seminorm Every seminorm induces a (pseudometrizable locally convex

• there are, are all TVS-isomorphic to one another.

Non-vector topologies

Discrete and cofinite topologies

is a non-trivial vector space (that is, of non-zero dimension) then the discrete topology (which is always metrizable) is 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 (where a subset is open if and only if its complement is finite) is also

Linear maps

of the origin.

A hyperplane is either dense or closed. A linear functional is continuous if and only if its kernel is 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, and the fact that the dual space of the space separates points in the space.

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

• F-spaces are complete topological vector spaces with a translation-invariant metric. These include L^p spaces

• 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

• 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 reflexivity condition, where the dual space is endowed with the topology of uniform convergence on totally bounded sets.

• Montel space: a barrelled space where every closed and bounded set is compact

• Fréchet spaces A locally convex F-space is a Fréchet space.

• LF-spaces are 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 or seminorm. In normed spaces a linear operator is continuous if and only if it is bounded.

• Banach spaces: Complete normed vector spaces of functions of bounded variation, and certain spaces of measures.

• Reflexive Banach spacess: 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 reflexive is L^1

• Hilbert spaces: these have an inner product Sobolev spaces and Hardy spaces.

• Euclidean spaces-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 of all continuous linear functionals, that is, continuous linear maps is continuous. This turns the dual into a locally convex topological vector space. This topology is called the 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 A topological vector space has a non-trivial continuous dual space if and only if it has a proper convex neighborhood of the origin.

Properties

the convex (resp. balanced, disked, closed convex, closed balanced, closed disked) hull The closure (respectively, interior, convex hull).

The convex hull is equal to the set of all which are finite linear combinations The intersection of any family of convex sets is convex and the convex hull of a subset is equal to the intersection of all convex sets that contain it.

Neighborhoods and open sets

Properties of neighborhoods and open sets

Every TVS is connected and locally connected and any connected open subset of a TVS is arcwise connected is a neighborhood of the origin.

and some positive continuous sublinear functional

is an absorbing disk is the Minkowski functional then where importantly, it was is a neighborhood of the origin).

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.

is a TVS that is of the second category in itself (that is, a nonmeager space) then any closed convex absorbing is a neighborhood of the origin. is not of the second category in itself.

Interior

The topological interior of a disk is not empty if and only if this interior contains the origin. is a balanced will necessarily be balanced; will be balanced if and only if it contains the origin. to also be convex (in addition to being balanced and having non-empty interior).; is not also convex;

then

and is a balanced (which are convex symmetric

Non-Hausdorff spaces and the closure of the origin

which is referred to as as a subset. The subspace topology is always the trivial topology a compact space (even if its dimension is non-zero or even infinite) and consequently also a bounded subset also carries the trivial topology and so is itself a compact, and thus also complete, subspace (see footnote for a proof). is not Hausdorff then there exist subsets that are both but ;

and this set is compact. Thus the closure of a compact subset of a TVS is compact (said differently, all compact sets are relatively compact), which is not guaranteed for arbitrary non-Hausdorff topological spaces.

(so that this open can be described as a "tube" the following are equivalent:

• is totally bounded.

• is totally bounded.

• is totally bounded.

• is totally bounded.

Moreover, the quotient map is always a closed map onto the (necessarily) Hausdorff TVS.

) is a topological complement has the indiscrete topology. is a Hausdorff completion

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. Thus, in a complete topological vector space, a closed and totally bounded subset is compact. is totally bounded is totally bounded, is totally bounded.

Every relatively compact set is totally bounded and the closure of a totally bounded set is totally bounded. The image of a totally bounded set under a uniformly continuous map (such as a continuous linear map for instance) is totally bounded. is totally bounded.

Closure and closed set

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

The closure of a vector subspace of a TVS is a vector subspace. Every finite dimensional vector subspace of a Hausdorff TVS is closed. The sum of a closed vector subspace and a finite-dimensional vector subspace is closed. 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 footnote for examples).

contain zero then

is convex.

then

is a convex neighborhood of the origin then equality holds.

and it is possible for this containment to be proper

Closed hulls

In a locally convex space, convex hulls of bounded sets are bounded. This is not true for TVSs in general.

• The closed disked

is compact then are compact) then

Hulls and compactness

In a general TVS, the closed convex hull of a compact set may to be compact. The balanced hull of a compact (respectively, totally bounded) set has that same property. The convex hull of a finite union of compact sets is again compact and convex.

Other properties

Meager, nowhere dense, and Baire

A 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.

is a TVS that does not carry the indiscrete topology is a Baire space has no balanced absorbing nowhere dense subset.

is nonmeager, which happens if and only if there does not exist a nowhere dense Every nonmeager locally convex TVS is a barrelled space.

Important algebraic facts and common misconceptions

is convex then equality holds. For an example where equality does also works.

The convex balanced hull is equal to the convex hull of the balanced hull where the inclusion might be strict since the balanced hull).

is a scalar then

Other properties

Every TVS topology can be generated by a of F-seminorms.

is some unary predicate).

Properties preserved by set operators

• The balanced hull of a compact (respectively, totally bounded, open) set has that same property.

• The (Minkowski) sum of two compact (respectively, bounded, balanced, convex) sets has that same property. But the sum of two closed sets need be closed.

• The convex hull of a balanced (resp. open) set is balanced (respectively, open). However, the convex hull of a closed set need be closed. And the convex hull of a bounded set need be bounded.

(indicated by the column name, "convex" for instance) is preserved under the set operator (indicated by the row's name, "closure" for instance). If in every TVS, a property is preserved under the indicated set operator then that cell will be colored green; otherwise, it will be colored red.

" 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

Proofs

Citations

Bibliography

Further reading

External links