Norm of Compact Hermitian Operator is Equal to Greatest Modulus of Eigenvalue

Theorem
Let $\struct {\HH, \innerprod \cdot \cdot_\HH}$ be a Hilbert space.

Let $T : \HH \to \HH$ be a compact Hermitian operator.

Then there exists a eigenvalue $\lambda$ of $T$ with:


 * $\cmod \lambda = \norm T$

where $\norm \cdot$ is the norm on the space of linear transformations.