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