# Definition:Topological Vector Space

Let $V$ be a vector space over a topological field $K$.
Let $\tau$ be a topology on $V$.
Then $\left({V, \tau}\right)$ is called a topological vector space if and only if:
 $(1)$ $:$ $\tau$ is a Hausdorff topology $(2)$ $:$ $+: V \times V \to V$ is continuous with respect to $\tau$ $(3)$ $:$ $\cdot: K \times V \to V$ is continuous with respect to $\tau$