An overview on the Prus-Szczepanik condition
Let \(N(X)\) be the family of all weakly null sequences in \(S(X)\), and \(M(X)\) a certain subclass. Let \[ d(\varepsilon,x)= \inf_{(y_m)\in N(X)} \limsup_{m\to\infty} \| x+\varepsilon y_m\|-\| x\|, \] \[ b(\varepsilon,x)= \sup_{(y_m)\in N(X)} \liminf_{ m\to\infty} \| x+\varepsilon y_m\|-\| x\| \] and \[ b_1(\varepsilon,x)= \sup_{(y_m)\in M(X)} \liminf_{ m\to\infty} \| x+\varepsilon y_m\|-\| x\|. \] A real Banach space without the Schur property is defined to be nearly uniformly noncreasy if for every \(\varepsilon>0\) there exists \(t>0\) such that for every \(x\) in the unit sphere \(S(X)\) it is true that \(d(\varepsilon,x)\geq t\) or \(b(t,x)\leq\varepsilon t\). The space \(X\) satisfies the Prus-Szczepanik (PSz) condition if there exists \(\varepsilon> 0\) such that, for every \(x\) in \(S(X)\), either \(b_1(1,x)< 1-\varepsilon\) or \(d(1, x)>\varepsilon\). The results of the paper arise from a study of the relationship between the above properties, the PSz condition and other well-known geometrical properties of Banach spaces. The interesting results are too involved to be quoted here.
scientific article; zbMATH DE number 6417859
scientific article