Definition 1

— Given a topological division ring $\mathbf{K}$ (GT, III, § 6.7) and a set $E$ such that $E$ has $1^\circ$ the structure of a left vector space on $\mathbf{K}$; $2^\circ$ a topology compatible with the structure of the additive group of $E$ (GT, III, § 1.1) and satisfying in addition the following axiom : (EVT) the mapping $(\lambda, x) \mapsto \lambda x$ of $\mathbf{K} \times E$ in $E$ is continuous, then $E$ is called a left topological vector space over (or on) $\mathbf{K}$.

It is equivalent to saying that $E$ is a topological left $\mathbf{K}\text{-module}$ (GT, III, § 6.6). A left vector space structure relative to $\mathbf{K}$ and a given topology on a set $E$, are said to be compatible if the topology and the additive group structure of $E$ are compatible and if, in addition, the axiom (EVT) is valid. This is the same as saying that the two mappings $(x, y) \mapsto x + y$ and $(\lambda, x) \mapsto \lambda x$ of $E \times E$ and of $\mathbf{K} \times E$, respectively, in $E$ are continuous, for then the mapping $x \mapsto -x = (-1)x$, is continuous and the topology of $E$ is compatible with its additive group structure.

If $E$ is a left topological vector space over $\mathbf{K}$, we say that $E$ provided only with its vector space structure, underlies the topological vector space $E$.