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A theorem of Corson and Lindenstrauss in:

Corson, H. H. and Lindenstrauss, J. “On weakly compact subsets of Banach spaces”. In: Proceedings of the American Mathematical Society 17.2 (1966), pp. 407–412.

http://www.ams.org/journals/proc/1966-017-02/S0002-9939-1966-0199669-9/S0002-9939-1966-0199669-9.pdf

states that:

Theorem: Let $(X, \lVert . \rVert)$ be a separable reflexive Banach space. A subset K of X is weakly compact if and only if it is the intersection of finite unions of cells.

in which a cell---also known as a ball---with center $x_0 \in X$ and radius $r \geq 0$ is defined as:

$$c(x_0,r) := \{ x \in X \mid \lVert x - x_0 \rVert \leq r\}.$$

In other words, a subset $K$ of $X$ is weakly compact if and only if it can be written as $K = \bigcap_{i \in I} U_i$, in which $I$ is an index set, and each $U_i$ is of the form $U_i = c^{(i)}_1 \cup c^{(i)}_2 \cup \cdots \cup c^{(i)}_{n_i}$, for some $n_i \in \mathbb{N}$, and some cells $c^{(i)}_1, c^{(i)}_2, \ldots, c^{(i)}_{n_i}$.

Note that the 'if' direction is kind of expected, and the stronger claim is the 'only if' part.

Now take the separable Hilbert space $X := \ell^2$ under its usual norm, and let $E := \{ e_n \mid n \in \mathbb{N} \}$ be the set of elements $e_n \in X$, defined as:

$$\forall k \in \mathbb{N} : e_n(k) := \left\{ \begin{array}{ll} 1, & \text{if }n = k,\\ 0, & \text{otherwise}.\end{array}\right.$$

We define the set $C$ as follows:

\begin{equation} C := \{ e_n \mid n \in \mathbb{N}\} \cup \{ 0 \}. \end{equation}

The set $C$ is sequentially weakly compact, hence by the Eberlein–Šmulian theorem, it is weakly compact (i.e., under the usual `finite open subcover' definition). Hence, by the Corson-Lindenstrauss thoerem, it can be represented as the intersection of finite unions of cells.

Question: Can anyone present an example of representing $C$ as the intersection of finite unions of cells?

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    $\begingroup$ I'm guessing that every weakly compact convex set is the intersection of cells. If so, then it is clear that for each finite set $E$ of $\mathbb{N}_0:=\mathbb{N}\cup\{0\}$, the set $(x_n)_{n\in E}\cup\overline{co}(x_n)_{n\notin E}$ the intersection of finite unions of cells, where $\overline{co}$ denotes "closed convex hull" and $(x_n)_{n=0}^\infty$ is formed from $x_0=0$ and $x_n=e_n$ for $n\in\mathbb{N}$. Now, is it true that for any $x\notin(x_n)_{n=0}^\infty$ we can find a finite subset $E(x)$ of $\mathbb{N}_0$ such that $x\notin\overline{co}(x_n)_{n\notin E(x)}$? Again I'm guessing yes. $\endgroup$
    – Ben W
    Aug 24, 2016 at 20:47
  • $\begingroup$ This is an interesting example. $\endgroup$
    – Amin
    Aug 24, 2016 at 23:18
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    $\begingroup$ It looks like the first condition (every weakly compact convex set is the intersection of cells) follows from the Mazur Intersection Property for $\ell_2$. (However it is apparently a deep result, and not true for arbitrary Banach spaces, although it is true for reflexive spaces with Frechet-differentiable norm.) I'm still thinking about the second condition. $\endgroup$
    – Ben W
    Aug 25, 2016 at 16:23
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    $\begingroup$ If $x \not\in (x_n)_{n=0}^{\infty}$ then in particular $x \neq 0$, and this implies that for some $k$, the $k$-th coordinate of $x$ is not zero, hence we can choose $E(x) := \{k\}$ (I think). So I think your example is correct. I'm just curious to know whether an explicit representation could be worked out. The proofs of Corson-Lindenstrauss or Mazur property do not immediately provide the explicit representation, at least my attempts have been futile. $\endgroup$
    – Amin
    Aug 25, 2016 at 22:03

1 Answer 1

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Let $x_n=-\frac{\sum_{k=1}^n e_k}{n}$. Then $\|e_i-x_n\|^2=\frac{n+3}{n}$ if $i \leq n$ and $\|e_i-x_n\|^2=\frac{n+1}{n}$ if $i>n$. So your set is the intersection of the sets $B(x_n,\sqrt{\frac{n+1}{n}}) \cup \bigcup_{i=1}^n B(e_i,\frac{1}{n})$.

Nice question, btw.

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