0
$\begingroup$

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.

$\endgroup$
6
  • $\begingroup$ 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
  • 1
    $\begingroup$ 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
1
$\begingroup$

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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