A remark on localized weak precompactness in Banach spaces (Q2761025)

Let \(X\) be a Banach space with dual \(X^*\) and let \(K\subset X^*\) be a weak{\(^*\)}-compact subset. A bounded set \(A\subset X\) is said to be a \(K\)-weakly precompact if every sequence \((x_n)\) has a subsequence \((x_{n_k})\) such that \((x^*,x_{n_k})\) converges for every \(x^*\in K\). NEWLINENEWLINENEWLINEThe author presents the following equivalent conditions: NEWLINENEWLINENEWLINE(i) \(A\) is \(K\)-weakly precompact. NEWLINENEWLINENEWLINE(ii) If \((x_n)\subset A\) and \(g\colon \overline {sp}\{x_n\mid n\geq 1\}\rightarrow \mathbb {R}\) is continuous convex function such that all subdifferentials \(\partial g(y)\) belong to \(\overline {co}^*(j^*K)\), where \(j^*\) is the dual map to inclusion \(j\colon \overline {sp}\{x_n\mid n\geq 1\}\rightarrow X\), then there is a dense \(G_\delta \) set \(G\subset X\) and a subsequence \((x_{n_k})\) of \((x_n)\) such that the directional derivatives \(Dg(y,x_{n_k})\) of \(g\) exist for all \(y\in \overline {sp}\{x_n\mid n\geq 1\}\) and are uniform with respect to the directions \((x_{n_k})\). NEWLINENEWLINENEWLINE(iii) If \((x_n)\subset A\) is a sequence and \(H\subset K\) non-empty, then there is a \(y\in \overline {sp}\{x_n\mid n\geq 1\}\) and a subsequence \((x_{n_k})\) of \((x_n)\) such that the directional derivatives \(Dg(y,x_{n_k})\) of the function \(g(x)=\sup \{(x^*,x)\mid x^*\in H\}\) at \(y\) exist uniformly in~\(x_{n_k}\).