Topological Dual Space

linear-algebra

Definition

Topological Dual Space

Let $(V,\Vert \cdot \Vert )$ be a normed vector space over $\mathbb{F}\in \{\mathbb{R},\mathbb{C}\}$. The topological dual space

$$V^{\ast }=\{f:V\to \mathbb{F}\mid f\text{ linear and continuous}\}$$

is the vector space of all continuous linear functionals on $V$, equipped with the operator norm

$$\Vert f{\Vert }_{V^{\ast }}=\underset{\Vert v\Vert \le 1}{\sup }|f(v)|.$$