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Let $X$ be a Banach space and $B_X$ be the unit ball.

Suppose that for each $\lbrace C_n\rbrace_{n=1}^\infty\subset B_X$ satisfying $C_n$ are closed convex and $C_n\supset C_{n+1}$ has a nonempty intersection.

Is it true that $X$ is reflexive?

3 Answers 3

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We will admit the following result:

A Banach space $X$ is reflexive if and only if for all $l:X\rightarrow \mathbb R$ linear and continuous we can find $x_0$ such that $\lVert x_0\rVert =\lVert l\rVert = \sup_{x\neq 0}\frac{l(x)}{\lVert x\rVert}$.

Let $l$ such a map. For all $n\in\mathbb{N}^*$, we can find $x_n$ with $\lVert x_n\rVert =1$ and $\langle l,x_n\rangle \geq \lVert l\rVert -\frac 1n$. Let $C_n$ the closed convex hull of the set $\left\{x_k,k\geq n\right\}$. $\left\{C_n\right\}$ is a decreasing sequence of closed convex non empty subsets of $B_X$. Let $x\in \bigcap_{n\in \mathbb{N}^*}C_n$. Let $\varepsilon>0$ and $k_0\in\mathbb N^*$. We can find $N\in\mathbb N$ and $k_1<\cdots0$ and $k_0>0$ we get that $\lVert l\rVert\lVert x\rVert \leq \langle l,x\rangle$ and it's an equality (we notice that $x\neq 0$, exect in the case in which $l=0$).

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    Very nice proof. My thoughts went exactly in the same direction. I remembered seeing the result about reflexivity in the book "Functional Analysis, Sobolev Spaces and Partial Differential Equations" by Haim Brezis, 2011 edition. It is a remark on page 4, and has some nice references for the proof of this result.2011-06-16
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    I have already red this result in the Haim Brezis' book. A reference given is "Geometry of Banach Spaces: selected topics" by Diestel.2011-06-16
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    Do you think the result you admitted is easier than the result to be proved?2011-06-16
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    @GEdgar: Indeed, this is the main problem of this proof. Since I don't have any other idea, I don't know if the result asked by @Jack Smith is easier. I will look at the reference given by @Zakk.2011-06-16
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    I didn't find Zakk's reference, but we answered Jack Smith's question.2011-06-17
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    @girdav: here's the original paper of [Smulyan](http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=sm&paperid=5784&option_lang=eng). It's in Russian but it contains an English summary at the end. Dieudonné has an article on (generalizations of) this work [here](http://dx.doi.org/10.1007/BF01900559). I think Holmes's book treats these results nicely. Essentially it's a criterion for weak compactness of bounded sets. Keywords: Smulyan, Eberlein-Smulyan.2011-06-21
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    @Theo Buehler Thanks for these references.2011-06-21
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Such $C_n$ exist in any Banach space. For example take $C_n$ to be the closed ball of radius $1-1/n$ centred at the origin.

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    Ok, I changed the question the original one doesn't make sense.2011-06-16
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It's true This is an old (1939) theorem due to Smulyan Look at any Banach space theory book for the proof