Let $H$ be a Hilbert space. Let $T \in L(H)$, i.e. $T$ is a linear continuous operator on $H$. Prove that if $T$ is an isomorphism, then $T^*$ is an isomorphism.
I have proved that $\ker T^{*}$ = $(\text{range}(T))^{\perp} = \{0\}$, and $(\text{range}(T^{*}))^{\perp} = \ker((T^{*})^*) = \ker T = \{0\}$
But I don't know how to prove $\text{range}(T^{*})$ is closed. Can someone help me with this? Or is there a better way to prove the statement?
I'm not saying you're approach won't work, but there is an easier way. If $T$ is an isomorphism, it has an inverse $T^{-1}$. Show that $(T^{-1})^*$ is the inverse of $T^*$.