$\|x\|_X \leq C \|Ax\|_Y,\quad C\gt 0$ Why is $A$ injective?

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.

1 Answer

If $Ax = 0$, then $0\le \|x\|_X\le C\|Ax\|_Y = 0$, so .....

• If $Ax=0$ then $x=0$, and hence the kernel is trivial and the operator is injective right – Functional Analysis Oct 24 '15 at 8:35
• Yes, you are right.@FunctionalAnalysis – user99914 Oct 24 '15 at 8:36
• And it being injective implies it has an inverse operator? – Functional Analysis Oct 24 '15 at 8:41
• No in general, it has to be onto.@FunctionalAnalysis – user99914 Oct 24 '15 at 9:01