Norm of Hermitian Operator

Theorem
Let $H$ be a Hilbert space.

Let $A \in B \left({H}\right)$ be a self-adjoint operator.

Then the norm of $A$ satisfies:


 * $\left\Vert{A}\right\Vert = \sup \left\{{ \left\vert{ \left\langle{Ah, h}\right\rangle_H }\right\vert: h \in H, \left\Vert{h}\right\Vert_H = 1 }\right\}$

Corollary
Suppose $A$ is self-adjoint and:


 * $\forall h \in H: \left\langle{Ah, h}\right\rangle_H = 0$

Then $A$ is the zero operator $\mathbf 0$.