Definition:Norm/Bounded Linear Transformation

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Definition

Let $\HH$ and $\KK$ be Hilbert spaces.

Let $A: \HH \to \KK$ be a bounded linear transformation.

Definition 1

The norm of $A$ is the real number defined and denoted as:

$\norm A = \sup \set {\norm {A h}_\KK: \norm h_\HH \le 1}$


Definition 2

The norm of $A$ is the real number defined and denoted as:

$\norm A = \sup \set {\dfrac {\norm {A h}_\KK} {\norm h_\HH}: h \in \HH, h \ne \mathbf 0_\HH}$


Definition 3

The norm of $A$ is the real number defined and denoted as:

$\norm A = \sup \set {\norm {A h}_\KK: \norm h_\HH = 1}$


Definition 4

The norm of $A$ is the real number defined and denoted as:

$\norm A = \inf \set {c > 0: \forall h \in \HH: \norm {A h}_\KK \le c \norm h_\HH}$


Also known as

The definition of a norm of a bounded linear transformation also applies when in fact $A$ is a linear operator (that is, $\HH = \KK$).

Hence the norm of a bounded linear operator is also defined.


As a case of pars pro toto, the norm defined here is commonly referred to as the operator norm, even when pertaining to a linear transformation.

However, in order not to cause confusion, that usage is deprecated on $\mathsf{Pr} \infty \mathsf{fWiki}$.


Also see


Sources