Definition:Topological Vector Space

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Definition

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 $\tau$