Topological Vector Space - Types of Topological Vector Spaces

Types of Topological Vector Spaces

Depending on the application additional constraints are usually enforced on the topological structure of the space. In fact, several principal results in functional analysis fail to hold in general for topological vector spaces: the closed graph theorem, the open mapping theorem, and the fact that the dual space of the space separates points in 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 Lp 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: these are locally convex spaces with the property that 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: Rn or Cn with the topology induced by the standard inner product. As pointed out in the preceding section, for a given finite n, there is only one n-dimensional topological vector space, up to isomorphism. It follows from this that any finite dimensional subspace of a TVS is closed. A characterization of finite dimensionality is that a Hausdorff TVS is locally compact if and only if it is finite dimensional (therefore isomorphic to some Euclidean space).