On infinite dimensional uniform smoothness of Banach spaces. (English) Zbl 1060.46504

Prus, Stanisław

Commentat. Math. Univ. Carol. 40, No. 1, 97-105 (1999).

The author studies a generalization of uniform smoothness as introduced in [T. Sekowski and A. Stachura, Atti. Semin. Mat. Fis. Univ. Modena 36, 329-338 (1988; Zbl 0681.46023)]. Uniformly smooth Banach spaces can be characterized as follows: \(X\) is uniformly smooth if for every \(\varepsilon >0\) there exists \(\delta >0\) so that for every \(x\) from the unit ball of \(X\), the diameter of the slice \(S(x,\delta )=\{ x^*\in B_{X^*}:\, x^*(x)\geq 1-\delta \}\) of the dual unit ball \(B_{X^*}\) is at most \(\epsilon \). A Banach space \(X\) is weakly nearly uniformly smooth (NUS\(^*\)), if for every \(\varepsilon >0\) there exists \(\delta >0\) so that for every \(x\) from the unit ball of \(X\) the Kuratowski measure of non-compactness of \(S(x,\delta )\) is at most \(\epsilon \).
Unlike uniformly smooth Banach spaces, NUS\(^*\) spaces do not have to be superreflexive, not even reflexive; \(c_0\) is an example. The author characterizes the NUS\(^*\) Banach spaces as those for which for every \(\varepsilon >0\) there is \(\eta >0\) so that if \(0<t<\eta \) and \((x_n)\) is a sequence in the unit ball of \(X\), then \(\| x_1+t(x_m-x_n)\| \leq 1+\varepsilon t\) for some \(n>m>1\). Using it, he for example shows that for every NUS\(^*\) Banach space there is \(p>1\) so that every shrinking finite-dimensional decomposition in \(X\) has a blocking which satisfies an upper \(\ell _p\) estimate. As a corollary, NUS\(^*\) Banach spaces have the weak Banach-Saks property and the weak fixed-point property.

Reviewer: Eva Matoušková (Praha)

MSC:

46B20	Geometry and structure of normed linear spaces
47H10	Fixed-point theorems

Keywords:

Banach space; nearly uniform smoothness; finite-dimensional decomposition; Banach-Saks property; fixed point property

Citations:

Zbl 0681.46023