Show that $T^{**}$ operator is symmetric for symmetric $T$

Question

I know that adjoint $T^*$ of the symmetric operator $T$ is not symmetric in general. However $T^{**}$ is the closure $\bar{T}$ and is hence symmetric. However is there a simpler way to see this property of $T^{**}$ without the fact that it forms closure of $T$ and without using graphs of $T$.

Answer 1

If $T$ is densely-defined and symmetric then $T\subseteq T^*$ so $T^{**}\subseteq T^*$ so $T^{**}\subseteq T^{***}$, therefore $T^{**}$ is symmetric. (${}^{*}$ is inclusion-reversing.)