The elements of topological vector spaces are typically functions, and the topology is often defined so as to capture a particular notion of convergence of sequences of functions.
A topological vector space X is a vector space over a topological field K (most often the real or complex numbers with their standard topologies) which is endowed with a topology such that vector addition X × X → X and scalar multiplication K × X → X are continuous functions.
Some authors require the topology on X to be Hausdorff, and some additionally require the topology on X to be locally convex (e.g., Fréchet space). For a topological vector space to be Hausdorff it suffices that the space be T1.
The category of topological vector spaces over a given topological field K is commonly denoted TVSK or TVectK. The objects are the topological vector spaces over K and the morphisms are the continuous K-linear maps from one object to another.
However, there are topological vector spaces whose topology does not arise from a norm, such as 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.
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 X of all functions f : R → R. X can be identified with the product space RR and carries a natural product topology. With this topology, X becomes a topological vector space, called the space of pointwise convergence. The reason for this name is the following: if (fn) is a sequence of elements in X, then fn has limit f in X if and only if fn(x) has limit f(x) for every real number x. This space is complete, but not normable.
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.
In particular, topological vector spaces are uniform spaces and one can thus talk about completeness, uniform convergence and uniform continuity. The vector space operations of addition and scalar multiplication are actually 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.
Vector addition and scalar multiplication are not only continuous but even homeomorphisms which means we can construct a base for the topology and thus reconstruct the whole topology of the space from any local base around the origin.
If a topological vector space is semi-metrisable, that is the topology can be given by a semi-metric, then the semi-metric must be translation invariant. Also, a topological vector space is metrizable if and only if it is Hausdorff and has a countable local base (i.e., a neighborhood base at the origin).
A linear function between two topological vector spaces which is continuous at one point is continuous on the whole domain.
If a vector space is finite dimensional, then there is a unique Hausdorff topology on it. Thus any finite dimensional topological vector space is isomorphic to Kn. A topological vector space is finite-dimensional if and only if it is locally compact. Here isomorphism means that there exists a linear homeomorphism between the two spaces.
Depending on the application we usually enforce additional constraints on the topological structure of the space. Below are some common topological vector spaces, roughly ordered by their niceness.
Every topological vector space has a continuous dual space—the set V* of all continuous linear functionals, i.e. continuous linear maps from the space into the base field K. A topology on the dual can be defined to be the coarsest topology such that the dual pairing V* × V → K 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 Banach space has a natural norm defined on it. However, it is very important in applications because of its compactness properties (see Banach-Alaoglu theorem).
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