Banach spaces $X,Y$ and a mapping $f:X\to Y$

Question

Let $X,Y$ be Banach spaces and a mapping $f\in \mathcal{L}(X,Y)$. Suppose that $f$ is also an injective map and an open map from $X$ to $Y$. Show that then $f \in \mathcal{B}(X,Y).$

Here I denoted the set of all linear mappings from $X$ to $Y$ by $\mathcal{L}(X,Y)$, and the set of all bounded mappings from $X$ to $Y$ by $\mathcal{B}(X,Y)$. Does anyone have an idea about this statement? I appreciate any hints.

Answer 1

$\textbf{Hint:}$ Using given conditions, show that $f$ is also surjective. Then appeal to the inverse mapping theorem.