topological vector space


A topological vector spaceMathworldPlanetmath is a pair (V,𝒯), where V is a vector spaceMathworldPlanetmath over a topological field K, and 𝒯 is a topologyMathworldPlanetmath on V such that under 𝒯 the scalar multiplication (λ,v)λv is a continuous functionPlanetmathPlanetmath K×VV and the vector addition (v,w)v+w is a continuous function V×VV, where K×V and V×V are given the respective product topologies.

We will also require that {0} is closed (which is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath to requiring the topology to be HausdorffPlanetmathPlanetmath), though some authors do not make this requirement. Many authors require that K be either or (with their usual topologies).

Topological vector spaces as topological groups

A topological vector space is necessarily a topological group: the definition ensures that the group operationMathworldPlanetmath (vector addition) is continuous, and the inverseMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath operationMathworldPlanetmath is the same as multiplication by -1, and so is also continuous.

Finite-dimensional topological vector spaces

A finite-dimensional vector space inherits a natural topology. For if V is a finite-dimensional vector space, then V is isomorphicPlanetmathPlanetmathPlanetmath to Kn for some n; then let f:VKn be such an isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmath, and suppose that Kn has the product topology. Give V the topology where a subset A of V is open in V if and only if f(A) is open in Kn. This topology is independent of the choice of isomorphism f, and is the finest ( topology on V that makes it into a topological vector space.

Title topological vector space
Canonical name TopologicalVectorSpace
Date of creation 2013-03-22 12:16:55
Last modified on 2013-03-22 12:16:55
Owner yark (2760)
Last modified by yark (2760)
Numerical id 22
Author yark (2760)
Entry type Definition
Classification msc 46A99
Synonym TVS
Synonym linear topological space
Synonym topological linear space
Related topic TopologicalRing
Related topic FrechetSpace