Definition:Topological Vector Space

Definition
Let $\mathcal{X}$ be a vector space over a field $K$ and $\tau$ a topology on $\mathcal{X}$.

$\left(\mathcal{X},\tau \right)$ is called a topological vector space if:


 * 1) $\tau$ is a Hausdorff topology.
 * $+:\mathcal{X}\times\mathcal{X}\to\mathcal{X}$ is continuous with respect to $\tau$.
 * 1) $\cdot:K\times\mathcal{X}\to\mathcal{X}$ is continuous with respect to $\tau$.