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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?

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    **Hint:** First find a sequence without convergent subsequence.2012-04-15
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    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)2012-04-15
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    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 $\sqrt{2}$.2012-04-15
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    @Matt: the collection of balls you start with doesn't cover the unit ball.2012-04-15
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    @t.b. I'm trying to see why they don't...2012-04-15
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    @Matt: e.g. $(-1,0,0,\ldots)$ is not contained in any of those balls.2012-04-15
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    @t.b. Right. But what if I add $(0, \dots, 0, -1, 0, \dots)$?2012-04-15

1 Answers 1

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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 $y_n(t) = \exp{(2\pi i \,n \,t)}$.

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    Using [Riesz's lemma](http://en.wikipedia.org/wiki/Riesz'_lemma) you can prove with the same idea that the unit ball in a normed space of infinite dimension is non-compact.2012-04-15
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    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 $V_N = X \smallsetminus \{y_n\,:\,n \geq N\}$.2012-04-15
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    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?2012-04-20
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    @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.2012-04-20
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    Ooh, I see. Thank you!2012-04-20