I saw the theorem 1.8 in the book Spectral Theory of ... that
If $T$ is a linear operator from the Hilbert space $H$ to another Hilbert space $H_1$, and the domain of definition of T, denoted by $\mathcal{D}(T)$, is $H$, then the adjoint of $T$, denoted by $T^*$, is bounded.
Here, the domain of definition of $T^*$, denoted by $\mathcal{D}(T^*)$, is the set of all $v\in H_1$ such that there exists $v^*\in H$ with $(Tu, v) = (u, v^*)$ for all $u\in \mathcal{D}(T)$.
There is no proof from the book. What I am trying is to use the closed graph theorem. First, we can show that $T^*$ is a closed linear operator. Next, to apply the closed graph theorem, we need $\mathcal{D}(T^*)$ be closed, but I can not show that.
Do you have any idea? Is the theorem true? How to prove it?