Let $c_0$ be the sequences with $\lim_{n\rightarrow \infty} = 0$. Show that the closed unit ball $\{x\in c_0, \|x\| \leq 1\}$ is not compact in $\ell^\infty$.

I know a lemma that says that the infinite closed unit ball are not compact in infinite-dimensional normed spaces.

This seems strange to me. Does not all the subsequences of sequences in $c_0$ converges?

  • 2
    $\begingroup$ I think you confuse elements of $l^\infty$ with sequences of elements of $l^\infty$. The point is that a subsequences of sequence of sequences (subsequences of sequence of elements of $l^\infty$) does not converge to any sequence (element of $l^\infty$). $\endgroup$ – dtldarek Dec 7 '12 at 7:02
  • $\begingroup$ Your title doesn't match your question. Your question also seems to be rather confused: compactness is a property of a space, and doesn't depend on any other space it might happen to be embedded in. $c_0$ is not compact, whether or not it happens to be embedded in $\ell^\infty$. $\endgroup$ – Chris Eagle Dec 7 '12 at 7:59
  • $\begingroup$ doesnt we need to consider the norm? $\endgroup$ – Johan Dec 7 '12 at 10:14

Let $e_n$ be $0$ except for a $1$ in the $n$th coordinate. Clearly $e_n \in c_0$. Then $\|e_n\|_\infty = 1$, but $\|e_n -e_m \|_\infty = 1$ whenever $n\neq m$. Hence $e_n$ can have no convergent subsequence, hence the set $\{ x \in c_0 | \|x\|_\infty \leq 1 \}$ cannot be compact.

And yes, all subsequences of $c_0$ must converge, since any subsequence of a convergent sequence must converge to the same limit.

  • $\begingroup$ thanks! I was thinking about this, but how do we know that this sequence are going on forever? and if its going on forever how can we say it is in $c_0$? $\endgroup$ – Johan Dec 7 '12 at 7:01
  • $\begingroup$ $e_n$ is zero except at index $n$. Hence $\lim_{i \to \infty} [e_n]_i = 0$. Hence $e_n \in c_0$. I do not understand what you mean by 'how do we know that this sequence are going on forever'. I have defined it for all $n$, so it must go on for all $n \in \mathbb{N}$. $\endgroup$ – copper.hat Dec 7 '12 at 7:20

Define $e_n$ to be the sequence that is zero everywhere except at $n$ where its one. Then $||e_n||=1$ and if we consider the sequence $\{e_n\}_{n=1}^\infty$ then I claim that it has no convergent subsequence. In particular notice that the sequence converges pointwise to $0$, so any limit would have to be $0$. But we have that $||e_n||=1$ so it can't converge in norm to $0$. Thereby it has no convergent subsequence.

Notice that the question is asking about a convergent subsequence of a sequence of sequences. Not a convergent subsequence of an element of $C_0$.


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