About Closed Graph Theorem for conjugate linear map

Question

Let $X$ be a Banach space over $\mathbb{C}$ and $T \colon X \to X$ be a conjugate linear map. If the graph $G(T)= \{ (x,Tx) \mid x\in X \}$ of $T$ is closed in $X\times X$, is $T$ continuous?

Answer 1

Hint: Let $Y$ be the 'conjugate' space of $X$, i.e. its underlying set and addition is that of $X$, and scalar multiplication is defined by conjugation $\lambda\bullet x:=\bar\lambda x$. Then use the closed graph theorem.