Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|improve this question
2  
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 $l^\infty$). –  dtldarek Dec 7 '12 at 7:02
    
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 $\ell^\infty$. –  Chris Eagle Dec 7 '12 at 7:59
    
doesnt we need to consider the norm? –  Johan Dec 7 '12 at 10:14

2 Answers 2

up vote 5 down vote accepted

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.

share|improve this answer
    
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 $c_0$? –  Johan Dec 7 '12 at 7:01
    
$e_n$ 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 $n \in \mathbb{N}$. –  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 $C_0$.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.