Definition

A topological vector space is a pair $(V,\mathcal{T})$ where $V$ is a vector space over a topological field $K$ and $\mathcal{T}$ is a topology on $V$ 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 $K$ 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 $f\colon V\to K^n$ 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 topology on $V$ that makes it into a topological vector space.