# Definition:Bounded Linear Functional

## Definition

### Normed Vector Space

Let $\mathbb F$ be a subfield of $\C$.

Let $\struct {X, \norm \cdot}$ be a normed vector space over $\mathbb F$.

Let $f : X \to \mathbb F$ be a linear functional.

We say that $f$ is a bounded linear functional if and only if:

there exists $C > 0$ such that $\cmod {\map f x} \le C \norm x$ for each $x \in X$.

### Inner Product Space

Let $\mathbb F$ be a subfield of $\C$.

Let $\struct {V, \innerprod \cdot \cdot}$ be an inner product space over $\mathbb F$.

Let $\norm \cdot$ be the inner product norm for $\struct {V, \innerprod \cdot \cdot}$.

Let $f : V \to \mathbb F$ be a linear functional.

We say that $f$ is a bounded linear functional if and only if:

there exists $C > 0$ such that $\cmod {\map f v} \le C \norm v$ for each $v \in V$.