Definition:Topological Dual Space
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
- Definition:Normed Dual Space - an instantiation of this concept in the case of $X$ a normed vector space.
Sources
- 1991: Walter Rudin: Functional Analysis (2nd ed.) ... (previous) ... (next): $1.6$: Topological vector spaces