Lemma. If $A$ is a bounded linear operator defined on a Hilbert space and $\|Af\| \geq c\|f\|$ and $\|A^*f\| \geq c\|f\|$ for some constant $c$. Then $A$ has a bounded inverse.
- In the proof of this lemma, it is claimed that $A$ and $A^*$ are both injective. I do not see why. Suppose that $Ax_1 = Ax_2$. Then we have $$0 = \|Ax_1 -Ax_2\| = \|A(x_1-x_2)\| \geq c\|x_1 -x_2\|.$$ Unless we have $c>0$, I do not see why $x_1 = x_2$? Am I missing something here, please?
- After the proof of injection, it is shown that $A$ is also onto and hence has an inverse. But how to see that this inverse is bounded, please? Thank you!
In the lemma, $c$ must be positive because for $A=0$ the statement clearly does not hold. For the 2nd question, use the open mapping theorem.