T attains its norm, then $T^*$ attains its norm

Question

Let $T:H\rightarrow H$ be a bounded linear operator. Suppose that $\exists$ $x \in H$. $||x||=1$ such that $||Tx||=||T||$. Show that $\exists$ $y \in H$, $||y||=1$ such that $||T^*y||=||T^*||$.

I tried using the formula that $||T||=sup_{||x||=1,||y||=1} |<Tx,y>|$. But I could not get any useful result this way.

Answer 1

Well using the formula $\exists x,y \in H$ such that $||T|| = |<Tx,y>|\implies ||T^*||=|<T^*y,x>| \leq ||T^*y||\leq ||T^*||$

This completes the solution