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:


 * $(1): \quad \tau$ is a Hausdorff topology
 * $(2): \quad +: \mathcal{X} \times \mathcal{X} \to \mathcal{X}$ is continuous with respect to $\tau$.
 * $(3): \quad \cdot: K \times \mathcal{X} \to \mathcal{X}$ is continuous with respect to $\tau$