Topological vector space

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 which is also a topological space, and where the vector space operations are continuous functions. This implies that the space has a uniform topological structure, allowing a notion of uniform convergence. Some authors also require that the space is a Hausdorff space.

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.

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:

Thus all Banach spaces and Hilbert spaces are examples of topological vector 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.

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.

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

A topological vector space embedding (abbreviated TVS embedding) or a topological monomorphism is an injective topological homomorphism. Equivalently, a TVS-embedding is a linear map that is also a topological embedding.[1]

A topological vector space isomorphism (abbreviated TVS isomorphism), also called a topological vector isomorphism [3] or an isomorphism in the category of TVSs, is a bijective linear homeomorphism. Equivalently, it is a surjective TVS embedding[1]

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

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

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.[4]

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 TVSs.

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

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.[10]

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

By the Birkhoff–Kakutani theorem, it follows that there is an equivalent metric that is translation-invariant.

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.

Every TVS is assumed to be endowed with this canonical uniformity, which makes all TVSs into uniform spaces. This allows one to[clarification needed] about 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.[16] A subspace 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).

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.

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.

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."

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.

In a locally convex space, convex hulls of bounded sets are bounded. This is not true for TVSs in general.[12]

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.[5] The convex hull of a finite union of compact convex sets is again compact and convex.[5]

A disk in a TVS is not nowhere dense if and only if its closure is a neighborhood of the origin.[8] A vector subspace of a TVS that is closed but not open is nowhere dense.[8]