If $T$ is an linear bijective isometry, is it true that $T^{-1}$ an isometry?

Question

Suppose that $X$ and $Y$ are Banach spaces. Let $T:X \rightarrow Y$ be a linear bijective isometry.

Question: Is it true that $T^{-1}$ also an isometry?

I think the answer is yes.

My attempt: We want to show that for any $y \in Y,$ we have $$\| T^{-1}y\| = \|y \|.$$ Since $T$ is an isometry, we have $$\| y \| = \| T T^{-1}y \| = \| T^{-1}y\|.$$

Is my proof correct?