Invertibility of a bounded linear operator

Question

Let $\mathcal{B}(F)$ be the algebra of all bounded linear operators on a complex Hilbert space $F$.

Is $T\in \mathcal{B}(F)$ and bijective, is $T$ invertible? i.e. is $T^{-1}\in \mathcal{B}(F)$?

Answer 1

If $T$ is bijective, then $T^{-1}\in \mathcal{B}(F)$. This is a consequence of the open mapping theorem.