Why is every selfadjoint operator closed?

I've read this theorem multiple times, but never seen a proof:

Every selfadjoint operator is closed.

But it's always been stated without a proof. Is it somehow obvious? I can't see it immediately from the graph.

Solution 1:

Let $T$ be a densely defined linear operator on a Hilbert space $H$. Recall that the adjoint $T^*$ is defined by the relation $$\langle T^*x,z\rangle=\langle x,Tz\rangle$$ – or more accurately $(x,y)$ are in the graph of $T^*$ if and only if $$\langle y,z\rangle=\langle x,Tz\rangle$$ for all $z$ in the domain of $T$. By the continuity of the maps $y\mapsto\langle y,z\rangle$ and $x\mapsto\langle x,Tz\rangle$, the set of all $(x,y)$ satisfying this relation is closed. Thus $T^*$ is a closed map.

If $T$ is selfadjoint then $T=T^*$, so $T$ itself is closed.