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?*



Corson-Lindenstrauss : Weakly compact sets as intersection of finite unions of cells
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.


