A closed set in Hilbert space does not have infimum norm

Question

Suppose $F$ is a closed set in Hilbert space. Can we construct such $F$ that $\inf_{x\in F}||x||$ cannot attain? My idea is to apply Sobolev norm on $C^1([0,1])$ space and show the incomplete. But I cannot figure out the exact procedure.

Thanks very much!

Answer 1

Take $H=l^2$, $$ F = \bigcup_n \left\{ (1+\frac1n)e_n\right\}. $$ Then $F$ is closed as it consists of isolated points, $\inf_{x\in F}\|x\|=1$ but $\|x\|>1$ for all $x\in F$.