# Let $H$ be a Hilbert space , If $T:H \to H$ has a bounded inverse $T^{-1}$ , then $T^{*}$ has a bounded inverse and $(T^*)^{-1}=(T^{-1})^*$

Let $$H$$ be a Hilbert space , If a continuous linear operator $$T:H \to H$$ has a bounded inverse $$T^{-1}$$ , then $$T^{*}$$ has a bounded inverse and $$(T^*)^{-1}=(T^{-1})^*$$

This theorem was in Michael Reed's functional analysis page $$186$$ .
I'm quite confused why we need to assume $$T$$ has a bounded inverse . If $$T$$ has an inverse $$T^{-1}$$ , then $$T$$ is necessarily bijective . In particular , $$T$$ is surjective . So by open mapping theorem on banach space , we conclude $$T$$ is an open mapping which means $$T^{-1}$$ is continuous . But on banach space , continuous is equivalent to bounded . So I don't understand why we need this assumption .

The assumptions make the proof easy. If $$TS=ST=I$$ where $$T,S$$ are bounded operators, then $$S^*T^*=T^*S^*=I^*=I$$ gives $$(T^*)^{-1}=S^*=(T^{-1})^*$$.