It is wellknown that there exists a homeomorphism $h$ from an infinitedimensional 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?
