Some geometric properties related to smoothness of Banach spaces
We introduce a new class of Banach spaces, locally \(k\) nearly uniformly smooth spaces, which is the dual notion to locally \(k\)-nearly uniformly convex spaces introduced by \textit{D.\,Kutzarova} [J.~Math.\ Anal.\ Appl.\ 162, No.\,2, 322--338 (1991; Zbl 0757.46026)]. The relationships between locally \(k\)-nearly uniformly smooth spaces, locally \(k\)-uniformly smooth Banach spaces, and locally fully \(k\)-smooth Banach spaces are discussed. We also introduce the two new smooth properties, (\(Sk\)-I) and (\(Sk\)-II), which are the extensions of smooth properties (S-I) and (S-II) that were introduced by \textit{Suyalatu} and \textit{X.--M.\ Jin} [J.~Anhui Norm.\ Univ., Nat.\ Sci.\ 22, No.\,1, 13--16 (1999; Zbl 0968.46018)], and discuss the relations between (\(Sk\)-I), (\(Sk\)-II) and other smooth properties. At last, the locally fully \(k\)-smoothness, compact locally fully \(k\)-smoothness, and the smooth properties (\(Sk\)-I) and (\(Sk\)-II) in product space \(l^p(X_i)\) are discussed.
scientific article; zbMATH DE number 5115293
scientific article