# Definition:Norm/Bounded Linear Functional

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*This page is about the norm on a bounded linear functional. For other uses, see Definition:Norm.*

## Contents

## Definition

Let $H$ be a Hilbert space, and let $L$ be a bounded linear functional on $H$.

### Definition 1

The **norm** of $L$ is the real number defined as the supremum:

- $\norm L = \sup \set {\size {L h}: \norm h_H \le 1}$

### Definition 2

The **norm** of $L$ is the supremum:

- $\norm{L} = \sup \set{\left|{Lh}\right|: \norm{h}_H = 1}$

### Definition 3

The **norm** of $L$ is the supremum:

- $\norm L = \displaystyle \sup \set {\frac {\size {L h} } {\norm h _H}: h \in H, h \ne \bszero_H}$

### Definition 4

The **norm** of $L$ is the infimum:

- $\norm{L} = \inf \set{c > 0: \forall h \in H: \left|{Lh}\right| \le c \norm{h}_H}$

As a consequence of definition $(4)$, we have for all $h \in H$ that $\size {L h} \le \norm L \norm h$.

As $L$ is bounded, it is assured that $\norm L < \infty$.

## Also see

### Special cases

- Definition:Hilbert Space
- Definition:Bounded Linear Functional
- Definition:Norm on Bounded Linear Transformation, of which this is a special case.

## Sources

- 1990: John B. Conway:
*A Course in Functional Analysis*... (previous) ... (next) $I.3.3$