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?

  • 2
    $\begingroup$ We can say that two norms $\|.\|_1$ and $\|. \|_2$ are equivalent if there is an isomorphism between $(X,\|.\|_1)$ and $(Y, \|.\|_2)$. The open mapping theorem show that a continuous bijective linear map has an inverse wich is continuous. You have to assume that $T$ is linear. $\endgroup$ – Patissot Mar 2 '15 at 17:38
  • $\begingroup$ @Patissot So you think the statement is not correct? $\endgroup$ – John Mar 2 '15 at 23:02

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