Let $X, Y$ and $Z$ be Banach spaces, and $M: X \rightarrow Y$ linear. Let also $S: Y\rightarrow Z$ be linear, injective and continuous, and the composition of operators $SM: X \rightarrow Z$ continuous. I need to show that $M$ is also continuous.
I know that if $M, S$ are continuous, then the composition is continuous, and the Operator norm holds, $\left\lVert SM\right\rVert \leq \left\lVert S\right\rVert \left\lVert M\right\rVert, $ where the norms are taken in the corresponding operator spaces. The reversal must not be true. I know that i need to show that $M$ is bounded.
Can somebody provide a proposal of proof ?