# Banach space and bounded below operators

Let $T\in L(X,Y)$, where $X, Y$ are Banach spaces. How can I conclude from $\ker(T) = \{0\}$ and $\mathrm{im}(T)$ closed that $T$ is bounded below, e.g. there exists a $c>0$ such that $||Tx||\geq c ||x||$?

$im(T)$ is a closed subspace of a banach space so itself is $banach$. $T:X\to im(T)$ is a surjective bounded linear operator between banach spaces, so open mapping theorem implies the result.