Continuous Linear Functional

linear-algebra

Definition

Continuous Linear Functional

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

A linear functional $f:V\to \mathbb{F}$ is continuous iff there exists $C\ge 0$ such that

$$|f(v)|\le C\Vert v\Vert \forall v\in V.$$

The smallest such $C$ is the operator norm

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