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It is well-known that there exists a homeomorphism $h$ from an infinite-dimensional Hilbert space $H$ to $H\setminus\{0\}$. Does there exist a "weak homeomorphism" $g:H \to H\setminus\{0\}$, that is, $g$ is bijective, and $g$ and $g^{-1}$ are weakly continuous?

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I don't get it: doesn't strong continuity imply weak continuity? so your $g$ could be your previous $h$? –  leo monsaingeon Jun 16 '13 at 7:17
Use the same topology on both ends. Then strong continuity need not imply weak continuity. –  Gerald Edgar Jun 16 '13 at 12:12
same quesion on MSE math.stackexchange.com/questions/421450/… –  Norbert Aug 19 '13 at 20:41

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