Why does $A:X\to Y$ where $X$ is a banach space and $Y$ is a normed space, where $A$ is a surjective bounded linear operator, where:
$$\|x\|_X \leq C \|Ax\|_Y,\quad C\gt 0$$
Mean that $A$ is also an injective operator?
I can't see it. Not sure where to start, it doesn't seem like it immediately jumps out at me.
If $Ax = 0$, then $0\le \|x\|_X\le C\|Ax\|_Y = 0$, so .....