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Topological vector space

In mathematics, a topological vector space X is a real or complex vector space which is endowed with a Hausdorff topology such that vector addition X × X -> X and scalar multiplication K × X -> X are continuous (where the product topologies are used and the base field K carries its standard topology).

Topological vector spaces are the most general spaces investigated in functional analysis. 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.

Table of contents
1 Examples
2 Types of topological vector spaces
3 Further properties


All normed vector spaces (and therefore all Banach spaces and Hilbert spaces) are examples of topological vector spaces. There are also many other examples however.

Consider for instance the set X of all functions f : RR. 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.

Here is another example: consider an open set D in Rn and the set X of infinitely differentiable functions f : DR. We first define a collection of semi-norms on X, and the topology will then be defined as the coarsest topology which refines the topology defined by each of the semi-norms. For a compact set K and a multi-index m = (m1, ..., mn) we define the (K, m) semi-norm to be the supremum of the differentiation first by x1 m1 times, then by x2 m2 times and so on K. With this topology, a sequence (fn) in X has limit f if and only if on every compact set all derivatives of fn converge uniformly to the corresponding derivative of f.

Types of topological vector spaces

We start the list with the most general classes and proceed to the "nicer" ones.

Further properties

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 a 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.

A topological vector space is finite-dimensional if and only if it is locally compact, in which case it is isomorphic to a Euclidean space Rn or Cn (in the sense that there exists a linear homeomorphism between the two spaces).

Every topological vector space has a dual - the set V* of all continuous linear functionals, i.e. continuous linear maps from the space into the base field K. The 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).