Topological vector space

In mathematics, a topological vector space 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 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. 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 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.
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:

The vector addition map defined by is continuous with respect to this topology. This follows directly from the triangle inequality obeyed by the norm.
The scalar multiplication map defined by where is the underlying scalar field of is 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 is a vector space over a topological field that is endowed with a topology such that vector addition and scalar multiplication are continuous functions. Such a topology is called a ' or a ' on
Every topological vector space is also a commutative topological group under addition.
Hausdorff assumption
Many authors, but not this page, require the topology on 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 of topological vector spaces over a given topological field is commonly denoted or The objects are the topological vector spaces over and the morphisms are the continuous -linear maps from one object to another.
A , also called a, is a continuous linear map between topological vector spaces such that the induced map is an open mapping when which is the range or image of is given the subspace topology induced by
A , also called a, is an injective topological homomorphism. Equivalently, a TVS-embedding is a linear map that is also a topological embedding.
A , 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
A collection of subsets of a vector space is called 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, to define a vector topology it suffices to define a neighborhood basis 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 ' of and for every index is called the -th knot of The set is called the beginning of The sequence is/is a:

' if for every index
Balanced if this is true of every
' if is summative, absorbing, and balanced.
' or a in a TVS if is a string and each of its knots is a neighborhood of the origin in

If is an absorbing 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 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.
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 is a collection sequences of subsets of then is said to be directed 'under inclusion or simply directed downward if is not empty and for all there exists some such that and .
Notation': Let be the set of all knots of all strings in
Defining vector topologies using collections of strings is particularly useful for defining classes of TVSs that are not necessarily locally convex.
If is the set of all topological strings in a TVS then 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 with respect to the operation of addition, and in a topological vector space the inverse operation is always continuous. Hence, every topological vector space is an abelian topological group. Every TVS is completely regular but a TVS need not be normal.
Let be a topological vector space. Given a subspace the quotient space with the usual quotient topology is a Hausdorff topological vector space if and only if is closed. This permits the following construction: given a topological vector space , form the quotient space where 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 :
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 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 : if for every there exists a real such that for any scalar satisfying
balanced or circled: if for every scalar
convex: if for every real
a disk or absolutely convex: if is convex and balanced.
symmetric: if or equivalently, if

Every neighborhood of the origin is an absorbing set and contains an open balanced neighborhood of so every topological vector space has a local base of absorbing and balanced sets. The origin even has a neighborhood basis consisting of closed balanced neighborhoods of if the space is locally convex then it also has a neighborhood basis consisting of closed convex balanced neighborhoods of the origin.
Bounded subsets
A subset of a topological vector space is bounded if for every neighborhood of the origin there exists such that.
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 the origin, 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 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.
Let be a non-discrete locally compact topological field, for example the real or complex numbers. A Hausdorff topological vector space over is locally compact if and only if it is finite-dimensional, that is, isomorphic to for some natural number