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 $\struct {V, \tau}$ 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 $\tau$      


Also see

  • Results about topological vector spaces can be found here.