topological vector space

Definition

A topological vector space is a pair $(V,𝒯)$,where $V$ is a vector space over a topological field $K$,and $𝒯$ is a topology on $V$ such that under $𝒯$the scalar multiplication $(\lambda ,v)\mapsto \lambda v$is a continuous function $K\times V\to V$and the vector addition $(v,w)\mapsto v+w$is a continuous function $V\times V\to V$,where $K\times V$ and $V\times V$ are given the respective product topologies.

We will also require that $\{0\}$ is closed(which is equivalent to requiring the topology to be Hausdorff),though some authors do not make this requirement.Many authors require that $K$ be either $ℝ$ or $ℂ$(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 operation (vector addition) is continuous,and the inverse operation 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 isomorphic to $K^n$ for some $n$;then let $f:V\to K^n$ be such an isomorphism,and suppose that $K^n$ 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 $K^n$.This topology is independent of the choice of isomorphism $f$,and is the finest (http://planetmath.org/Coarser) topology on $V$that makes it into a topological vector space.