Bolzano-Weierstrass theorem holds in any NLS. (Normed Linear Space). Is this statement True?

If it is true how to prove it.

I know how to prove it in $\mathbb R$. I proved it in $\mathbb R$ by using supremum infimum of the set. But in NLS supremum infimum concept do not exist.

Can anyone please help me by giving any hint.

  • $\begingroup$ Are you sure this is true? This question or rather the answers, seem to show the opposite. Or do you mean for you space to be finite-dimensional? $\endgroup$ – saulspatz Sep 8 '18 at 3:54
  • $\begingroup$ math.stackexchange.com/q/2293062/489079 please check this link $\endgroup$ – cmi Sep 8 '18 at 4:06
  • $\begingroup$ This is not true in general normed spaces, unless you specify that the sequence is contained in a compact set. $\endgroup$ – Rodrigo Dias Sep 8 '18 at 4:12
  • $\begingroup$ Can you please give a counter example?@rldias $\endgroup$ – cmi Sep 8 '18 at 4:15
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    $\begingroup$ Bolzano - Wierstrass holds in a normed linear space iff it is finite dimensional. In other words, every infinite dimensional normed linear space has a closed and bounded set which is not compact. $\endgroup$ – Kavi Rama Murthy Sep 8 '18 at 12:12

Consider the space $l_1$ and the sequence $e_n$ (the $n$th unit vector). This is bounded but had no convergent subsequence.


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