# Definition:Topological Vector Space

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## 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** if and only if:

\((1)\) | $:$ | $+_X: \struct {X \times X, \tau_X \times \tau_X} \to \struct {X, \tau_X}$ is continuous | |||||||

\((2)\) | $:$ | $\circ_X : \struct {K \times X, \tau_K \times \tau_X} \to \struct {X, \tau_X}$ is continuous |

## Also defined as

Many sources require that topological vector spaces be Hausdorff.

On $\mathsf{Pr} \infty \mathsf{fWiki}$, 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.

## Also see

- Results about
**topological vector spaces**can be found**here**.

## Sources

- 1963: John L. Kelly and Isaac Namioka:
*Linear Topological Spaces*(2nd ed.) ... (previous) ... (next): $5$: Linear topological spaces, linear functionals, quotient and products