Does the isometry group of a *real* separable infinite-dimensional Hilbert space have two connected components? Or, conversely, is the there even a Kuiper's theorem in the real case?


How does the isometry group of a real infinite-dimensional Hilbert separable look like?



Isometry Group of real Hilbert space?
It is contractible, according to [Kuiper himself](http://www.ams.org/mathscinet-getitem?mr=179792) who also writes:



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> Putnam and Wintner [[5](http://www.ams.org/mathscinet-getitem?mr=41848), [6](http://www.ams.org/mathscinet-getitem?mr=45121)] proved with the help of spectral resolutions that $U\_{\mathbf R}$ and hence $GL\_{\mathbf R}$ are connected. 
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