Definition:Norm/Bounded Linear Transformation
Definition
Let $\GF \in \set {\R, \C}$.
Let $\struct {X, \norm {\, \cdot \,}_X}$ and $\struct {Y, \norm {\, \cdot \,}_Y}$ be normed vector spaces over $\GF$.
Let $A: X \to Y$ be a bounded linear transformation.
Definition 1
The norm of $A$ is the real number defined and denoted as:
- $\norm A = \sup \set {\norm {A x}_Y : \norm x_X \le 1}$
Definition 2
The norm of $A$ is the real number defined and denoted as:
- $\norm A = \sup \set {\dfrac {\norm {A x}_Y} {\norm x_X}: x \in X, x \ne \mathbf 0_X}$
This supremum is to be taken in $\closedint 0 \infty$ so that $\sup \O = 0$.
Definition 3
The norm of $A$ is the real number defined and denoted as:
- $\norm A = \sup \set {\norm {A x}_X : \norm x_X = 1}$
This supremum is to be taken in $\closedint 0 \infty$ so that $\sup \O = 0$.
Definition 4
The norm of $A$ is the real number defined and denoted as:
- $\norm A = \inf \set {c > 0: \forall x \in X: \norm {A x}_Y \le c \norm x_X}$
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, $X = Y$).
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
- Definition:Hilbert Space
- Definition:Bounded Linear Transformation
- Definition:Norm on Bounded Linear Functional
- Norm on Bounded Linear Transformation is Finite
- Fundamental Property of Norm on Bounded Linear Transformation
- Norm on Bounded Linear Transformation is Submultiplicative
- Definition:Operator Norm: the term used for a norm when $\HH = \KK$
Sources
- 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) $\S \text {II}.1$