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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 $4.3$.

Idonknow
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    $W$ is non-separable and it's being mapped under a linear map, so its image is non-separable. It is also weakly compact, and the image of a compact set under a continuous operator is compact: $T$ being cts with respect to the norm topologies imply that $T$ is cts with respect to the weak topologies. – James Kilbane Nov 06 '15 at 11:46
  • I understand why $T(W)$ is weakly compact. So you are saying that linear map preserves separability? – Idonknow Nov 06 '15 at 16:13

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