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, when $X$ is a normed vector space, $X^\ast$ will always denote the topological dual the given norm on $X$ (that is, the normed dual space).

Consequently, the topological dual 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.