On infinite dimensional generalization of Yan's theorem (Q2708137)

The generalization mentioned in the title reads as follows. Denote by \(B\) the set of all elements \(\{f_i\}\in L^1 (\ell^2; \mu)\) with \(f_i\geq 0\) and \(f_i\in L^\infty\) (\(\mu\) is a probability measure). Let \(K\) be a convex subset of \(L^1 (\ell^2)\) such that \(0\in K\). The following conditions are equivalent: NEWLINENEWLINENEWLINE(a) for every f in the positive cone of \(L^1(\ell^2)\) there exists \(\lambda >0\) such that \(\lambda f\notin \overline{K-B}\), NEWLINENEWLINENEWLINE(b) for every set \(A\) of positive measure there exists \(\lambda > 0\) such that \(\lambda\chi_A e_i\notin\overline{K-B}\) (\(e_i\) is the \(i\)th basic unit vector of \(\ell^2\)), NEWLINENEWLINENEWLINE(c) there exists \(Z\in L^\infty (\ell^2)\) such that \(P\{\omega: (Z(\omega), e_i)>0\} =1\) for \(i=1, 2,\ldots\), and \(\sup_{Y\in K}{\mathbf E}[(Z, Y)]<\infty\). NEWLINENEWLINENEWLINEThe original Yan's theorem is the same about the scalar \(L^1\) [see \textit{J.-A. Yan}, Lect. Notes. Math. 784, 220-222 (1980; Zbl 0429.60004)].