As stated in the title, let be a $T $ a continuous, linear, open map from a normed space $X$ onto a normed space $Y $. I would like to know if it is true that there exists a constant $C$ such that for any $x \in X, \ \ ||x||_X \le C||Tx||_Y $.
I tried to verify it but got stuck. Is the claim true?
Of course, if such a constant exists, we must have $C>0$. And $T$ must be injective; otherwise, there will be a non-zero vector $x$ such that $Tx=0$ and so $\lVert x\rVert>C\lVert Tx\rVert=0$.
On the other hand, suppose now that $T$ is injective. Since $T$ is open and linear it is surjective. So, since $T$ is injective too, it has an inverse. Let $C=\lVert T^{-1}\rVert$. Then$$\lVert x\rVert=\bigl\lVert T^{-1}(Tx)\bigr\rVert\leqslant C\lVert Tx\rVert.$$
So, such a constant $C$ exits if and only if $T$ is injective.
