Definition:Topological Dual Space

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Definition

Let $K$ be a topological field.

Let $\struct {X, \tau}$ be a topological vector space over $K$.

Let $\struct {X, \tau}^\ast$ be the vector space of continuous linear functionals on $\struct {X, \tau}$.


We say that $\struct {X, \tau}^\ast$ is the topological dual space of $\struct {X, \tau}$.


Notation

Where the topology $\tau$ is clear from context, we write $X^\ast$ for $\struct {X, \tau}^\ast$ and say that $X^\ast$ is the topological dual (space) of $X$.

On $\mathsf{Pr} \infty \mathsf{fWiki}$, when $X$ is a normed vector space, $X^\ast$ will always denote the topological dual with respect to the given norm on $X$ (that is, the normed dual space).

Consequently, the topological dual with respect to any other topology will use the full notation $\struct {X, \tau}^\ast$.


Also see


Sources