Topological Vector Space - Topological Structure

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.

Let X be a topological vector space. Given a subspace M ⊂ X, the quotient space X/M with the usual quotient topology is a Hausdorff topological vector space if and only if M is closed. This permits the following construction: given a topological vector space X (that is probably not Hausdorff), form the quotient space X / M where M is the closure of {0}. X / M is then a Hausdorff vector topological space that can be studied instead of X.

In particular, topological vector spaces are uniform spaces and one can thus talk about completeness, uniform convergence and uniform continuity. (This implies that every Hausdorff topological vector space is completely regular.) 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 said to be normable if its topology can be induced by a norm. A topological vector space is normable if and only if it is Hausdorff and has a convex bounded neighbourhood of 0.

If a topological vector space is semi-metrizable, that is the topology can be given by a semi-metric, then the semi-metric can be chosen to 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 operator between two topological vector spaces which is continuous at one point is continuous on the whole domain. Moreover, a linear operator f is continuous if f(V) is bounded for some neighborhood V of 0.

A hyperplane on a topological vector space X is either dense or closed. A linear functional f on a topological vector space X has either dense or closed kernel. Moreover, f is continuous if and only if its kernel is closed.

Every Hausdorff finite dimensional topological vector space is isomorphic to Kn for some topological field K. In particular, a Hausdorff topological vector space is finite-dimensional if and only if it is locally compact.