Lifting of rotundity properties from \(E\) to \(L^ p(\mu,E)\)
A Banach space \(E\) is said to be weakly uniformly rotund (weakly locally uniformly rotund) if \(\| x_ n\| =\| y_ n\| =1\) for every \(n\in {\mathbb{N}}\) and \(\| x_ n+y_ n\| \to^{w}2\) imply that \(x_ n-y_ n\to^{w}0\) (resp \(\| x\| =\| y_ n\| =1\) for every \(n\in {\mathbb{N}}\) and \(\| x+y_ n\| \to 2\) imply that \(y_ n\to^{w}x)\). Moreover, the conjugate Banach space \(B^*\) of \(B\) is said to be weakly* uniformly rotund (weakly* locally uniformly rotund) if \(\| f_ n\| =\| g_ n\| =1\) for every \(n\in {\mathbb{N}}\) and \(\| f_ n+g_ n\| \to 2\) imply that \((f_ n+g_ n)\to^{w^*}0\) (resp. \(\| f\| =\| g_ n\| =1\) for every \(n\in {\mathbb{N}}\) and \(\| f+g_ n\| \to 2\) imply that \(g_ n\to^{w^*}f).\) It is shown that \(L^ p(\mu,E)\) with \(1<p<\infty\) (the Lebesgue-Bochner function space of \(\mu\)-equivalence classes of strongly measurable functions \(f:S\to E\) with \(\int_{S}\| f(s)\|^ pd\mu <\infty\) for a given measure space \((S,\Sigma,\mu)\)) is weakly uniformly rotund (weakly locally uniformly rotund) if and only \(E\) is weakly uniformly rotund (resp. weakly locally uniformly rotund) and \((L^ p(\mu,E))^*\) with \(1<p<\infty\) is weakly* uniformly rotund (weakly* locally uniformly rotund) if and only if \(E^*\) is \(weakly^*\) uniformly rotund (resp. weakly* locally uniformly rotund).
scientific article; zbMATH DE number 4032375