# Norm of Self-Adjoint Operator

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## Contents

## 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 also that:

- $\forall h \in H: \sequence {A h, h}_H = 0$

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

## Proof

## Sources

- 1990: John B. Conway:
*A Course in Functional Analysis*... (previous) ... (next) $II.2.13, 14$