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 $||Tx||\geq c ||x||$?
Banach space and bounded below operators
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functional-analysis