Definition:Norm/Bounded Linear Transformation

Definition

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

Then the norm of $A$, denoted $\norm A$, is the real number defined by:

$(1): \quad \norm A = \sup \set {\norm {A h}_K: \norm h_H \le 1}$
$(2): \quad \norm A = \sup \set {\dfrac {\norm {A h}_K} {\norm h_H}: h \in H, h \ne \mathbf 0_H}$
$(3): \quad \norm A = \sup \set {\norm {A h}_K: \norm h_H \le 1}$
$(4): \quad \norm A = \inf \set {c > 0: \forall h \in H: \norm {A h}_K \le c \norm h_H}$

These definitions are equivalent, as proved in Equivalence of Definitions of Norm of Linear Transformation.

An important property of $\norm A$ is that:

$\forall h \in H: \norm {A h}_K \le \norm A \norm h_H$

This is proven on Submultiplicativity of Operator Norm.

As $A$ is bounded, it is assured that $\norm A < \infty$.

Operator Norm

Above definition 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.