Where can I read about the Stone-Cech compactification of the separable Hilbert space?
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Hilbert space $l^2$ is homeomorphic to product space $\mathbb R^\infty$, but Stone-Cech compactification of that space is not very nice either. Probably the best thing is to read about the Stone-Cech compactification in general. Check (haha) your library for The Stone-Cech Compactification by Russel Walker.