# Bounded inverse operator

Let $X$ and $Y$ be Banach spaces. Suppose $T$ is a linear operator from $X$ onto $Y$ with $\operatorname{Dom}(T)\subset X$. Show that $\exists T^{-1}\in L(Y,X)\Leftrightarrow\exists M>0:\ \left\Vert x\right\Vert \leq M\left\Vert Tx\right\Vert ,\forall x\in \operatorname{Dom}(T)$ .

This problem seems to be closely related to the following theorems. But I also noticed some differences. For example, it is possible that $\operatorname{Dom}(T)\neq X$. Also, we don't know $T\in L(X,Y)$. Thank you!

$T\in L(X,Y)$, $\operatorname{Dom}(T)=X$. Then, $\operatorname{Ker}(T)=\{0\}$ (one-to-one) and $\operatorname{Im}(T)$ is closed in $Y$ iff $\left\Vert Tx\right\Vert _{Y}\geq C\left\Vert x\right\Vert _{X}$ $\forall x\in X$ for some $C>0$.

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