Definition:Norm/Bounded Linear Transformation

This page is about the norm on a bounded linear transformation (the operator norm). For other uses, see Definition:Norm.

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:

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

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

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

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

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 (i.e., $H = K$).

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.