Suppose that $\| \cdot \|$ and $|||\cdot|||$ are two norms on a vector space $X$ such that $|||x||| \leq C \|x\|$ for some $C > 0$ and all $x \in X$. Let $T$ be the identity operator from $(X,\| \cdot \|)$ to $(X, ||| \cdot|||)$.
I am trying to show that $T$ is a quotient operator if and only if it is an isomorphism, but am struggling with both directions. By quotient operator I mean that $T$ is surjective and that the map $T_0: X/Ker T \to Ran T$ given by $T_0(x + Ker T)= Tx$ is an isometric isomorphism.
My thoughts: the condition $|||x||| \leq C \|x\|$ gives that $T$ is bounded, and it is clearly a bijection. I imagine the rest is a bit of a definition chase, but I can not get anywhere with it. I would appreciate any help!
I think I've figured it out. It can be proven by making use of the fact that $T^{-1}$ is bounded if and only if there exists $r>0$ such that $\|Tx\| \geq r\|x\|$.