How do I exhibit an open cover of the closed unit ball of the following:

(a) $X = \ell^2$
(b) $X=C[0,1]$
(c) $X= L^2[0,1]$

that has no finite subcover?

  • 4
    $\begingroup$ Hint: First find a sequence without convergent subsequence. $\endgroup$ – t.b. Apr 15 '12 at 20:09
  • $\begingroup$ I was going to expand on this ^ comment when his answer appeared but since I use different words, I'll post what I was going to write anyway: For a), if you take $(0, \dots, 1, 0 , \dots )$ the sequences that are $1$ at position $k$ and zero otherwise then they're all in the closed unit ball in $\ell^2$. Take your open cover to be the $\varepsilon$-balls around these sequences and $\varepsilon < \sqrt{2}$. (to be continued) $\endgroup$ – Rudy the Reindeer Apr 15 '12 at 20:26
  • $\begingroup$ Then pick any finite subcover of this. Then there will be an $N$ such that all the sequences in your cover are zero after $N$. Pick a point $(0, \dots, 0, 1, 0, \dots )$ where $1$ appears somewhere above $N$. Then this point is not covered by your finite subcover since its distance to any of the sequences in your basis is $\sqrt{2}$. $\endgroup$ – Rudy the Reindeer Apr 15 '12 at 20:29
  • $\begingroup$ @Matt: the collection of balls you start with doesn't cover the unit ball. $\endgroup$ – t.b. Apr 15 '12 at 20:30
  • $\begingroup$ @t.b. I'm trying to see why they don't... $\endgroup$ – Rudy the Reindeer Apr 15 '12 at 20:35

Suppose $\{y_n\}_{n=1}^\infty$ is a sequence in the closed unit ball $B$ which has no accumulation point. Then $U_{N} = B \smallsetminus \{y_n\,:\,n \geq N\}$ is open and $B \subset \bigcup_{N=1}^\infty U_N$. However, $\{U_{N}\}_{N=1}^\infty$ has no finite subcover.

For a) take the standard basis, for b) take $y_{n}(t) = t^n$ and for c) try $y_n(t) = \exp{(2\pi i \,n \,t)}$.

  • 1
    $\begingroup$ Using Riesz's lemma you can prove with the same idea that the unit ball in a normed space of infinite dimension is non-compact. $\endgroup$ – t.b. Apr 15 '12 at 20:40
  • $\begingroup$ It was brought to my attention that it may be confusing that $U_N$ is open in $B$ but not in the surrounding space $X$. If this is the case just replace $U_N$ by $V_N = X \smallsetminus \{y_n\,:\,n \geq N\}$. $\endgroup$ – t.b. Apr 15 '12 at 20:51
  • $\begingroup$ I'm sorry for this question: there are two proofs, one with Riesz and then the one you've given above, is that right? Your comment sounds as if I should do the same thing with Riesz but the thing seems to work just fine without him. Maybe I'm just confused? $\endgroup$ – Rudy the Reindeer Apr 20 '12 at 18:49
  • $\begingroup$ @MattN. In these specific instances ($\ell^2$, $C[0,1]$ and $L^2[0,1]$) you can exhibit a sequence without accumulation points directly. In an abstract infinite dimensional Banach space you need to argue that such a sequence exists. Riesz's lemma allows you to do that. $\endgroup$ – t.b. Apr 20 '12 at 18:55
  • $\begingroup$ Ooh, I see. Thank you! $\endgroup$ – Rudy the Reindeer Apr 20 '12 at 18:57

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