The limit of best generalized peak norm approximations (Q5955059)

Let \(X\) be a Banach space of measurable functions on a measure space \((X,\Sigma,\mu)\) without atoms and with \(\mu(X)= 1\). Set \(\|f\|^{(\alpha)}:= \sup_{\mu(A)=\alpha} \|f\chi_A\|\), \(f\in F\), \(0< \alpha\leq 1\). In particular, \(\|f\|^{(1)}=\|f\|\). For \(U\subset X\) one defines \(d_\alpha(f;U):= \inf_{u\in U}\|f-u\|^{(\alpha)}\) and \(P_\alpha(f; U):= \{u\in U: d_\alpha(f; U)=\|f-u\|^{(\alpha)}\}\). The authors establish conditions that provide NEWLINE\[NEWLINE\lim_{\alpha\to 1-0} d_\alpha(f,U)= d(f,U)\quad (=d_1(x,U))NEWLINE\]NEWLINE and \(\lim_{\alpha\to 1-0} P_\alpha(f; U)= P(f;U)\). The same problem is studied for \(\alpha\to +0\).