Hellinger-Toeplitz Theorem

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

Let $T : \HH \to \HH$ be a symmetric linear operator.

That is:


 * $\innerprod {T x} y = \innerprod x {T y}$ for each $x, y \in \HH$.

Then:


 * $T$ is bounded.