Definition:Norm/Bounded Linear Transformation

Definition
Let $H, K$ be Hilbert spaces, and let $A: H \to K$ be a bounded linear transformation.

Operator Norm
The definition of a norm of a bounded linear transformation also applies when in fact $A$ is a linear operator (that is, $H = K$).

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

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

Also see

 * Definition:Hilbert Space
 * Definition:Bounded Linear Transformation
 * Definition:Norm on Bounded Linear Functional, a special case where $K$ is in fact the ground field of $H$.
 * Operator Norm is Finite where it is shown that the norm is assured to be finite
 * Submultiplicity of Operator Norm where it is shown that:
 * $\forall h \in H: \norm {A h}_K \le \norm A \norm h_H$


 * Equivalence of Definitions of Norm of Linear Transformation where it is shown these definitions are equivalent