Lemma 7

Let $u$ be a closed symmetric operator on a complex Hilbert space E and $v$ a closed symmetric extension of $u$. The domain of $u$ is a closed subspace of the Hilbert space $E_v$.

Since $v$ is an extension of $u$, we have $(x|y)_v= (x|y)_u$ for $x$ and $y$ in dom($u$). The canonical injection of $E_u$ into $E_v$ is therefore isometric, and the conclusion follows (lemma 8 of I, p. 107).