Definition:Norm/Bounded Linear Functional

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 $\norm L < \infty$.