Assume $\Gamma$ acts by isometries on a separable Hilbert space $H$, and 
$$\operatorname{diam} H/\Gamma\le 1.$$ 
Is it true that $H/\Gamma$ is compact?

 

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**Stupid example.** Assume the action of $\Gamma$ on $H=\ell_2$ is generated by coordinate translations $x_n\mapsto x_n+\epsilon_n$. Then 
$$\operatorname{diam} H/\Gamma=\tfrac12\cdot\sqrt{\sum_{n=1}^\infty\epsilon_n^2}.$$ 
Thus, if $\operatorname{diam} H/\Gamma\le 1$ then $H/\Gamma$ is a quotient of Hilbert cube, and has to be compact.