We have just introduced linear operators in my FA class. Let $X,Y$ be normed spaces and $T:X \rightarrow Y$ a linear operator.
Claim: $T$ has a continuous inverse $T^{-1}$ on $T(X)$, if and only if there exists $c>0$ such that $$c \|x\|_X \le \|Tx\|_Y, \text{ for all } x\in X.$$
I have the typical introductory lemma and theorems at my disposal.
Ideas: Under the assumption that such an inverse exists, I first noted that $T$ is surjective on $T(X)$, with the intention of utilizing that $T^{-1}$ would be linear, if $T$ was bijective. However, I couldn't conclude conclude anything about injectivity.
If I could show that $T$ was continuous, I'd know that for some $C\ge0$ holds $\|Tx\|_Y \le C\|x\|_X$, which in turn I could maybe use to come somewhere near my claim. I'm not sure how to tackle this problem. I don't know if $T$ is continuous, which seems to be my biggest problem for now.
Am I missing something, which would allow me to conclude bijectivity of $T$, linearity of $T^{-1}$ or continuity of $T$? If not, what else is noteworth, which could be utilized to proove the claim?
Best Answer
If we are given the existence of a continuous inverse, then $$ c\|T^{-1}y\|_X \leq \|y\|_Y \ \text{for all } y \in T(X) \iff c\|x\|_X \leq \|Tx\|_Y \ \text{for all } x \in X, $$ so $T$ is indeed bounded away from zero. On the other hand, suppose that $T$ is bounded away from zero. Define the map $S:T(X) \to X$ by $S(T(x)) = x$ so that $S$ is the inverse of $T$. We have $$ c\|x\|_X \leq \|Tx\|_Y \ \text{for all } x \in X \iff c\|Sy\|_X \leq \|y\|_Y \ \text{for all } y \in T(X), $$ so the resulting map $S$ is bounded.