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 which is also a topological space, the latter thereby admitting a notion of continuity.
More specifically, its topological space has a uniform topological structure, allowing a notion of uniform convergence.
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.
Hilbert spaces and Banach spaces are well-known examples.
Unless stated otherwise, the underlying field of a topological vector space is assumed to be either the complex numbers ℂ or the real numbers ℝ.

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.
Some authors require the topology on to be T1; it then follows that the space is Hausdorff, and even Tychonoff.
The topological and linear algebraic structures can be tied together even more closely with additional assumptions, the most common of which are listed below.
The category of topological vector spaces over a given topological field is commonly denoted TVS? or TVect?.
The objects are the topological vector spaces over and the morphisms are the continuous ?-linear maps from one object to another.

Ways of defining a vector topology

If is a vector space and is a topology on, then the addition map is continuous at the origin if and only if the set of neighborhoods of 0 in is additive.
This is consequently a necessary condition for a topology to form a vector topology.
Since every vector topology is translation invariant, to define a vector topology it suffices to define a neighborhood basis for it at the origin.

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:
  1. The vector addition is jointly continuous with respect to this topology. This follows directly from the triangle inequality obeyed by the norm.
  2. The scalar multiplication, where is the underlying scalar field of, is jointly continuous. This follows from the triangle inequality and homogeneity of the norm.
Therefore, 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.
If is a vector space then the trivial topology is a TVS topology on.
If is a non-trivial vector space then the discrete topology on is not a TVS topology; furthermore, the cofinite topology on is also not a TVS topology on.

Finest vector topology

Let be a real or complex vector space.
There exists a TVS topololgy on that is finer than every other TVS-topology on .
Every linear map from into another TVS is necessarily continuous.
If has an uncountable Hamel basis then is not locally convex and not 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.
For instance, the set of all functions : this set can be identified with the product space and carries a natural product topology.
With this topology, becomes a topological vector space, endowed with a topology called the topology of pointwise convergence.
The reason for this name is the following: if is a sequence of elements in, then has limit if and only if has limit for every real number x.
This space is complete, but not normable: indeed, every neighborhood of 0 in the product topology contains lines, i.e., sets for.

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.
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 M is closed.
This permits the following construction: given a topological vector space , form the quotient space where M is the closure of.
is then a Hausdorff topological vector space that can be studied instead of.
In particular, topological vector spaces are uniform spaces and one can thus talk about completeness, uniform convergence and uniform continuity.

The vector space operation of addition is uniformly continuous and the scalar multiplication is Cauchy continuous.
Because of this, every topological vector space can be completed and is thus a dense linear subspace of a complete topological vector space.
The Birkhoff–Kakutani theorem gives that the following three conditions on a topological vector space are equivalent:
A metric linear space means a vector space together with a metric for which addition and scalar multiplication are continuous.
By the Birkhoff–Kakutani theorem, it follows that there is an equivalent metric that is translation-invariant.
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 0.
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 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 is closed.
Let be a non-discrete locally compact topological field, for example the real or complex numbers.
A topological vector space over is locally compact if and only if it is finite-dimensional, that is, isomorphic to for some natural number.

Local notions

A subset of a topological vector space is said to be