Proposition 10 (Hellinger–Toeplitz)

Let $u$ be a symmetric partial operator on the Hilbert space E. If the domain of $u$ is equal to E, then $u\in \mathscr{L}(E)$ and $u$ is hermitian.

In fact, the partial operator $u$ is closable (prop. 8 of IV, p. 237), and therefore closed since its domain is E (IV, p. 228, Remark). We then conclude by invoking EVT, I, p. 19, cor. 5.