Definition

A topological vector space is a pair $(V,\mathcal{T})$ , where is a vector space over a topological field , and $\mathcal{T}$ is a topology on such that under $\mathcal{T}$ 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 be either $\mathbb{R}$ or $\mathbb{C}$ (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 , and so is also continuous.

Finite-dimensional topological vector spaces

A finite-dimensional vector space inherits a natural topology. For if is a finite-dimensional vector space, then is isomorphic to for some ; then let $f\colon V\to K^n$ be such an isomorphism, and suppose that has the product topology. Give the topology where a subset of is open in if and only if is open in . This topology is independent of the choice of isomorphism , and is the finest topology on that makes it into a topological vector space.