Definition:Norm (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 $\left\|{A}\right\|$, is the real number defined by:

$(1): \qquad \left\Vert{A}\right\Vert = \sup \left\{{\left\Vert{Ah}\right\Vert_K: \left\Vert{h}\right\Vert_H \le 1}\right\}$

$(2): \qquad \left\Vert{A}\right\Vert = \sup \left\{{\dfrac {\left\Vert{Ah}\right\Vert_K} {\left\Vert{h}\right\Vert_H}: h \in H, h \ne \mathbf{0}_H}\right\}$

$(3): \qquad \left\Vert{A}\right\Vert = \sup \left\{{\left\Vert{Ah}\right\Vert_K: \left\Vert{h}\right\Vert_H \le 1}\right\}$

$(4): \qquad \left\Vert{A}\right\Vert = \inf \left\{{c > 0: \forall h \in H: \left\Vert{Ah}\right\Vert_K \le c \left\Vert{h}\right\Vert_H}\right\}$

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

An important property of $\left\|{A}\right\|$ is that:

$\forall h \in H: \left\Vert{Ah}\right\Vert_K \le \left\Vert{A}\right\Vert \left\Vert{h}\right\Vert_H$

This is proven on Submultiplicativity of Operator Norm.

As $A$ is bounded, it is assured that $\left\Vert{A}\right\Vert < \infty$.

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.

See also

• Norm (Linear Functional), a special case where $K$ is in fact the ground field of $H$.

Sources

• John B. Conway: A Course in Functional Analysis (1990) $\S II.1$