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.

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.

Also see

 * Equivalence of Definitions of Norm of Linear Transformation


 * 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 $\HH = \KK$