I just learnt open mapping theorem. And I met a statement online asserting that
If $X$,$Y$ are Banach space, and $T:X\to Y$ be a continuous bijection, then norms for $X,Y$ are equivalent.
Can we define equivalence of norms between two different spaces? And since the map may not be linear, continuity may not imply boundedness, and open mapping theorem can't apply. How should I prove the statement?