TITLE: Can the surface of the closed unit ball be compact
QUESTION [9 upvotes]: It is known that in the Banach space with infinite dimension, the closed unit ball is not compact. I am just wondering if it is possible for the surface of the ball, say $\|x\|=1$ can be compact.

REPLY [22 votes]: No, because then the closed unit ball would be compact, since it is the image of the map$$\begin{array}{ccc}[0,1]\times\{x\,|\,\|x\|=1\}&\longrightarrow&\text{your space}\\(t,x)&\mapsto&tx.\end{array}$$