Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

If $X$ is a reflexive Banach space and $(C_n), n \in \mathbb{N}$ is a sequence of closed convex bounded sets with $C_{n+1}$ contained in $C_n$ for all $n \in \mathbb{N}$. How does one show that the countable intersection of $C_n$ for $n \in \mathbb{N}$ is not the empty set?

share|cite|improve this question
How does one mark some answers as correct? – nada Jun 10 '12 at 18:43

Hint: Do you know the Eberlein-Shmulyan theorem?

share|cite|improve this answer
Would it be possible to please give me a few more details? I am not too aware of how to use it to prove the statement. – nada Jun 10 '12 at 18:45
I need to prove that X being reflexive is weak sequentially compact(by the Eberlein Shmulyan theorem). But how does that follow from the statement given in the question? – nada Jun 10 '12 at 20:15
Also, I have seen many versions of the theorem on the internet. Which one is applicable here? – nada Jun 10 '12 at 20:19
If $x_n \in C_n$, Eberlein-Shmulyan says some subsequence has a weak limit point $x$. Show that $x$ is in all the $C_n$. – Robert Israel Jun 10 '12 at 22:00

the proof can be seen in Introduction to Banach Space Theory by Robert E. Megginson.

share|cite|improve this answer
Please try to describe as much here as possible in order to make the answer self-contained. – robjohn Apr 30 '13 at 13:29

Your Answer


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.