The book I am using for my Introduction of Topology course is Principles of Topology by Fred H. Croom.

Show that Hilbert Space is not locally compact at any point.

This is what I understand:

  • A space $X$ is compact provided that every open cover of $X$ has a finite subcover.
  • A space $X$ is locally compact at a point $x$ in $X$ provided that there is an open set $U$ containing $x$ for which $\overline{U}$ is compact. $X$ is locally compact provided that it is locally compact at each point.
  • A metric space is compact if and only if it has the Bolzano-Weierstrass property.
  • Local compactness does not imply compactness.

My rough attempt:

Seeking to prove Hilbert Space $H$ is not locally compact at any point by contradiction. Suppose $H$ is locally compact at a point $p = (x_1,x_2,...)$. Let $U$ be an open set containing $p$. Since $H$ is locally compact, $\overline{U}$ is compact; thus $\exists r>0:B(p, r)\subset U$. Then $\overline{B(p,r)} = B[p,r] \subset \overline{U}$. However, the set $P= \{p_n\}_{n=1}^{\infty}$ of points $p_n= (x_1,x_2,...,x_{n-1},x_n+r/2,x_{n+1},...)$ is an infinite subset of $B[p,r]$ with no limit point. Since compactness is equivalent to the Bolzano-Weierstrass property in metric spaces, we must conclude that $B[p,r]$ is not compact. Thus $\overline{U}$ is not compact and $H$ is not locally compact at any point.

Is there anything I need to change regarding the proof? Any suggestions? Anything I need to clarify?

I sincerely thank you for taking the time to read this question and my attempt at proving it. I greatly appreciate any assistance you may provide.

  • 1
    $\begingroup$ You would want your Hilbert space to be infinite-dimensional. Closed balls in a finite-dimensional Hilbert space are compact by the Heine-Borel Theorem. $\endgroup$ – Berrick Caleb Fillmore Apr 20 '15 at 0:37
  • $\begingroup$ I think it may be more illuminating to prove that a normed space is locally compact if and only if it is finite dimensional. $\endgroup$ – JessicaK Apr 20 '15 at 0:40
  • $\begingroup$ "contradiction. Suppose" $\: \mapsto \:$ "contradiction, suppose" $\;\;\;\;$ $\endgroup$ – user57159 Apr 20 '15 at 1:08

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