Creating a reflexive Banach space from a weakly compact set. I have a set $K$ which is weakly compact in a Banach space $E$.  Also, its span is norm-dense in $E$.  ($E$ is weakly compactly generated.  But this doesn't enter the question anywhere.)
I need to construct a reflexive Banach space, so I need its unit ball to be compact.  Since $K$ is compact, I thought I might try to use the space $\operatorname{span}(K)$.  I want its unit ball to still be weakly compact, is this the case?
That is, if $K$ is weakly compact in $E$, then is $B_{Y}$ weakly compact in $Y$, where $Y = \operatorname{span}(K)$?
 A: I think the answer is no. Let $E=\ell^1(\mathbb{N})$ and let  $\{e_n\}_{n\in\mathbb{N}}$ be the canonical basis. Then put
$$
K=\left\{\frac1n\,e_n: \ n\in\mathbb{N}\right\}\cup\{0\}.
$$
Then $K$ is even norm-compact (because it is itself a norm-convergent sequence), but the closure of the span of $K$ is all of $\ell^1(\mathbb{N})$.
The same trick can be used on any separable Banach space, many of which are not reflexive and thus have non-weak-compact balls. 
A: I am going to suggest a different answer to the accepted one, but I hope it may be of interest to someone. In particular, Corollary $3$ below may be noteworthy. In this answer we shall need to use the Krein-Smulian Weak Compactness Theorem, which says the following: the closed convex hull of a weakly compact subset of a Banach space is itself weakly compact (the original reference for this result is M. Krein and V. Smulian, On regularly convex sets in the space conjugate to a Banach space, Ann. Math. (second series) 41 (1940) pp.556-583, but proofs abound.)
The idea will be to take the absolutely convex hull of $K$ (that is, the convex hull of the  weakly compact set $K\cup (-K)$), and form a Banach space from this set; this idea comes from the paper Factoring Weakly Compact Operators, J. Funct. Anal. 17 (1974) pp.311–327 by Davis, Figiel, Johnson and Pelczynski. I won't give all the details from their paper, but I will quote below several parts of the text of their paper:

Let $W$ be a bounded, convex, symmetric subset of a Banach space
  $(X,\,\Vert \cdot\Vert)$. For each $n=1, 2,\ldots$ the gauge
  $\Vert\cdot\Vert_n$ of the set $U_n=2^nW+2^{-n}B_X$ is a norm
  equivalent to $\Vert\cdot\Vert$. Define, for each $x\in X$,
  $\vert\vert\vert\, x \, \vert\vert\vert= (\sum_{n=1}^\infty\Vert
 x\Vert_n^2)^{1/2}$, let $Y= \{ x\in X \mid \vert\vert\vert\, x \,
 \vert\vert\vert<\infty\}$ and $C=B_Y = \{ x\in X \mid
 \vert\vert\vert\, x \, \vert\vert\vert\leq 1\}$. Finally, let $j$
  denote the identity embedding of $Y$ into $X$.
$\vdots$
Here are some properties of this construction:
Lemma 1.
(i) $W\subseteq C$.
(ii) $(Y,\, \vert\vert\vert\cdot\vert\vert\vert)$ is a Banach space and $j$ is continuous.
(iii) $j^{\ast\ast}:Y^{\ast\ast}\longrightarrow X^{\ast\ast}$ is one-to-one and $(j^{\ast\ast})^{-1}(X)=Y$.
(iv) $Y$ is reflexive if and only if $W$ is weakly relatively compact.

Note that in part (iii) of Lemma $1$ above, $X$ and $Y$ are considered as subsets of their respective biduals in the usual way. I will not reproduce the proof of Lemma $1$ here, but proofs of the following three corollaries of Lemma $1$ are brief enough that it is easy for me to reproduce them here (I shall add a little information in the proof of the third corollary.)

Corollary 1. Weakly compact operators factor through reflexive spaces.
Proof. Let $T:Z\longrightarrow X$ be weakly compact and let $W$ of Lemma $1$ be $T(B_Z)$. The operators $j^{-1}\circ T:Z\longrightarrow Y$ and $j: Y\longrightarrow X$ provide the required factorization.     $\square$
Corollary 2. Every weakly compact subset $K$ of a Banach space $X$ is affinity homeomorphic (in the respective weak topologies) to a subset of a reflexive Banach space.
Proof. In Lemma $1$ let $W$ be the convex hull of $K\cup (-K)$. Then, by the Krein-Smulian Weak Compactness Theorem, $W$ is weakly relatively compact, hence, by (iv) above, $Y$ is reflexive. Therefore $K'=j^{-1}(K)$ is weakly relatively compact (being weakly closed and bounded, by (i).) Note that $j|_{K'}$ is the homeomorphism we need.     $\square$
Corollary 3. A Banach space $X$ is weakly compact generated if and only if there is a reflexive Banach space $R$ and a one-to-one operator $T:R\longrightarrow X$ with $T(R)$ dense in $X$.
Proof. The sufficiency is obvious. For the necessity, let $K$ be a weakly compact subset of $X$ whose linear span is norm dense in $X$ and in Lemma $1$ let $W$ be the convex hull of $K\cup (-K)$, so that $Y$ is reflexive and $j$ gives the desired one-to-one operator.    $\square$

