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

Examples

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 : R → R. X can be identified with the product space R^R 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 (f_n) is a sequence of elements in X, then f_n has limit f in X if and only if f_n(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 : D → R. 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, ..., m_n) 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 (f_n) in X has limit f if and only if on every compact set all derivatives of f_n 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.

• Locally convex topological vector spaces: here each point has a local base consisting of convex sets. By a technique known as Minkowski functionals it can be shown that a space is locally convex if and only if its topology can be defined by a family of semi-norms. Local convexity is the minimum requirement for "geometrical" arguments.
• Fréchet spaces: these are complete locally convex spaces where the topology comes from a translation-invariant metric, or equivalently: from a countable family of semi-norms. Many interesting spaces of functions fall into this class.
• Banach spaces: Complete normed vector spaces. Most of functional analysis is formulated for Banach spaces.
• Reflexive Banach spacess: Banach spaces naturally isomorphic to their double dual (see below), which ensures that some geometrical arguments can be carried out. An important example which is not reflexive is L¹, whose dual is L^∞ but is strictly contained in the dual of L^∞.
• Hilbert spaces: these have an inner product; even though these spaces may be infinite dimensional, most geometrical reasoning familiar from finite dimensions can be carried out in them.

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).