Definition:Norm (Linear Functional)

This page is about the norm on bounded linear functionals. For other uses, see Definition:Norm.

Definition

Let $H$ be a Hilbert space, and let $L$ be a bounded linear functional on $H$.

Then the norm of $L$, denoted $\left\|{L}\right\|$, is the real number defined by:

$(1): \qquad \left\|{L}\right\| = \sup \left\{{\left|{Lh}\right|: \left\|{h}\right\| \le 1}\right\}$

$(2): \qquad \left\|{L}\right\| = \sup \left\{{\left|{Lh}\right|: \left\|{h}\right\| = 1}\right\}$

$(3): \qquad \left\|{L}\right\| = \displaystyle \sup \left\{{\frac {\left|{Lh}\right|} {\left\|{h}\right\|}: h \in H, h \ne \mathbf{0}_H}\right\}$

$(4): \qquad \left\|{L}\right\| = \inf \left\{{c > 0: \forall h \in H: \left|{Lh}\right| \le c \left\|{h}\right\|}\right\}$

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

As a consequence of definition $(4)$, have for all $h \in H$ that $\left\vert{Lh}\right\vert \le \left\Vert{L}\right\Vert \left\Vert{h}\right\Vert$.

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

See also

• Norm (Linear Transformation), of which this is a special case.

Sources

• John B. Conway: A Course in Functional Analysis (1990) $I.3.2$