Definition:Topological Vector Space

Definition
Let $\struct {K, +_K, \circ_K, \tau_K}$ be a topological field.

Let $\struct {X, +_X, \circ_X, \tau_X}$ be a vector space over $K$.

Let $\tau_X \times \tau_X$ be the product topology of $\tau_X$ and $\tau_X$.

Let $\tau_K \times \tau_X$ be the product topology of $\tau_K$ and $\tau_X$.

We say that $\struct {X, \tau_X}$ is called a topological vector space :

Also defined as
Many sources require that topological vector spaces be Hausdorff.

On, in the interest of presenting results in their fullest generality, we do not do this.

Results that require $\struct {X, \tau}$ to be Hausdorff should use Definition:Hausdorff Topological Vector Space instead.