An estimate for the type of a complexly uniformly convex Banach space
Let X be a complex Banach space with the unit sphere S. The modulus of convexity \(\delta\) (t) of X is introduced by \(\delta (t)=\inf \{\max \{\| x+thy\|:h\in H\}:x,y\in S\}-1\), \(t>0\), where \(H=\{1,-1,i,- i\}\). According to \textit{J. Globevnik} [Proc. Am. Math. Soc. 47, 175-178 (1975; Zbl 0307.46015)] the space X is called complexly uniformly convex if \(\delta (t)>0\) for all \(t>0\). The class of such Banach spaces is substantially larger than the class of ordinary uniformly convex spaces; for instance, \(L_ 1\) is complexly uniformly convex. The author shows that if X is complexly uniformly convex, then it has some cotype in the following sense: There are \(q\geq 2\) and \(K>0\) such that \(\max \{\| h_ 1x_ 1+...+h_ nx_ n\|^ q:\) \(h_ 1,...,h_ n\in H\}\geq K(\| x_ 1\|^ q+...+\| x_ n\|^ q)\) for all \(x_ 1,...,x_ n\in X\) and all \(n=1,2,... \). The proposed proof works also for ordinary uniformly convex spaces, thus reproving in a simple way the well known results. [Reviewer's remark. The result of this paper is contained in Section 8 of the paper of \textit{W. J. Davis, D. J. H. Garling} and \textit{N. Tomczak- Jaegermann}, J. Funct. Anal. 55, 110-150 (1984; Zbl 0552.46012)].
scientific article; zbMATH DE number 3969368