Suppose that $X$ is a nonseparable weakly compactly generated Banach space. Let $W$ be a weakly compact subset which spans a dense linear subspace of $X$.
Denote $\mathcal{F}(X) = \overline{span\{ \delta_x : x \in X \}}$ where $\delta_x$ is an evaluation funtional on $Lip(X)$, given by $\delta_x(f)=f(x)$ for $f \in Lip(X)$ (set of Lipschitz functions).
If $\beta_X : \mathcal{F}(X) \rightarrow X$ and $T : X \rightarrow \mathcal{F}(X)$ such that $T$ is a continuous linear map which satisfies $\| T \| =1$ and $\beta_X T = id_X$, then $T(W)$ is a weakly compact nonseparable subset of $\mathcal{F}(X)$.
How to prove the statement above? Any hint?
Remark: The statement above is taken from here, Theorem $4.3$.