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On locally uniformly rotund Banach spaces. (Spanish. English summary) Zbl 0785.46017

A Banach space \((X,\|\cdot\|)\) is said to be locally uniformly rotund (LUR) if for every \(x_ 0\in S_ X\) and every \(\varepsilon>0\) there exists \(\delta=\delta(\varepsilon,x_ 0)>0\) such that \(\| x_ 0-y\|\geq \varepsilon\) and \(y\in S_ X\) imples \(\| {{x_ 0+y} \over 2}\|\leq 1-\delta\) where \(S_ X\) is the unit sphere in \(X\). Let \(p\) satisfy \(1<p<\infty\) and let \(\{X_ n\): \(n\in\mathbb{N}\}\) be a countable family of Banach spaces. Let \(\ell^ p(X_ 1,X_ 2,\dots)\) with the norm \(\|\cdot\|_ p\) be defined as \(\{x=(x_ n)\in \prod_{n=1}^ \infty X_ n\): \(\| x\|_ p^ p= \sum_{n=1}^ \infty \| x_ n\|^ p<\infty\}\). The authors prove that the space \(\ell^ p(X_ 1,X_ 2,\dots)\) is (LUR) if and only if every space \(X_ n\), \(n=1,2,\dots\) is also (LUR).

MSC:

46B20 Geometry and structure of normed linear spaces
46B45 Banach sequence spaces
46E40 Spaces of vector- and operator-valued functions
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