Question

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

Answer 1

• A topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space.A space Xis locally compact at a point x in X provided that there is an open set U containing x for which U  is compact. X is locally compact provided that it is locally compact at each point.

Proof:

Seeking to prove Hilbert Space H is not locally compact at any point by contradiction. Suppose H

 is locally compact at a point p=(x₁,x_2,....)Let $\overline{U}$  be an open set containing p. Since H

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 to infinity} 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.