Equivalence of Definitions of Norm of Linear Functional

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

Then the following definitions of the norm of $L$ are equivalent:


 * $(1): \qquad \left\|{L}\right\| = \sup \left\{{\left|{Lh}\right|: \left\|{h}\right\| \le 1}\right\}$
 * $(2): \qquad \left\|{L}\right\| = \sup \left\{{\left|{Lh}\right|: \left\|{h}\right\| = 1}\right\}$
 * $(3): \qquad \left\|{L}\right\| = \displaystyle \sup \left\{{\frac {\left|{Lh}\right|} {\left\|{h}\right\|}: h \in H, h \ne \mathbf 0}\right\}$
 * $(4): \qquad \left\|{L}\right\| = \inf \left\{{c > 0: \forall h \in H: \left|{Lh}\right| \le c \left\|{h}\right\|}\right\}$

Corollary
For all $h \in H$, the following inequality holds:


 * $\left|{Lh}\right| \le \left\|{L}\right\| \left\|{h}\right\|$