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?*