Topological vector space

In mathematics a topological vector space is one of the basic structures investigated in functional analysis. As the name suggests the space blends a topological structure (a uniform structure to be precise) with the algebraic concept of a vector space.

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.

Hilbert spaces and Banach spaces are well-known examples.

Definition

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.

N.B. Though we do not do so here, 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). However, it is known that for a topological vector space to be Hausdorff it suffices that the space is a T₁ space.

The category of topological vector spaces over a given topological field K is commonly denoted TVS_K or TVect_K. The objects are the topological vector spaces over K and the morphisms are the continuous K-linear maps from one object to another.

Examples

All normed vector spaces (and therefore all Banach spaces and Hilbert spaces) are examples of topological vector spaces.

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 X of all functions f : R → R. X can be identified with the product space R² 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. This space is complete, but not normable.

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 (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.

Every topological vector space has a local base of absorbing and balanced sets.

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.

A linear functional f on a topological vector space X is continuous if and only if kernel(f) is closed in X.

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 Hausdorff 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.

Types of topological vector 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.

• 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 like the Hahn-Banach theorem.
• Barrelled spaces: locally convex spaces where the Banach-Steinhaus theorem holds.
• Montel space: a barrelled space where every closed and bounded set is compact
• Bornological space: a locally convex space where the continuous linear operators to any locally convex space are exactly the bounded linear operators.
• LF-spaces are limits of Fréchet spaces. ILH spaces are inverse limits of Hilbert spaces.
• F-spaces are complete topological vector spaces with a translation-invariant metric. These include L^p spaces for all p > 0.
• 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. A locally convex F-space is a Fréchet space.
• Nuclear spaces: a kind of Fréchet space where every bounded map from the nuclear space to an arbitrary Banach space is a nuclear operator.
• Normed spaces and semi-normed spaces: locally convex spaces where the topology can be described by a single norm or semi-norm. In normed spaces a linear operator is continuous if and only if it is bounded.
• Banach spaces: Complete normed vector spaces. Most of functional analysis is formulated for Banach spaces.
• Reflexive Banach spaces: 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.
• Euclidean spaces: these are finite dimensional Hilbert spaces. According to above comments, any locally compact Hausdorff TVS is isomorphic (as a topological vector space) to one Euclidean space.

Dual space

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).

References

• A Grothendieck: Topological vector spaces, Gordon and Breach Science Publishers, New York, 1973. ISBN 0-677-30020-4
• G Köthe: Topological vector spaces. Grundlehren der mathematischen Wissenschaften, Band 159, Springer-Verlag, New York, 1969.
• Schaefer, Helmuth H. (1971). Topological vector spaces. New York: Springer-Verlag. ISBN 0-387-98726-6.
• Lang, Serge (1972). Differential manifolds. Reading, Mass.–London–Don Mills, Ont.: Addison-Wesley Publishing Co., Inc.
• F Trèves: Topological Vector Spaces, Distributions, and Kernels, Academic Press, 1967. ISBN 0-486-45352-9.