Weakly compactly LUR Banach spaces
Variants of the well-known concept of local uniform rotundity (LUR, in short) and of its weak version WLUR (use weak convergence instead of norm convergence in the definition) are considered in the literature on the geometry of the closed unit ball \(B_X\) of a Banach space \(X\). The space \(X\) is said to be compactly locally uniformly rotund (CLUR, in short) if \(\{x_n\}\) has a norm-convergent subsequence whenever \(x, x_n\in S_X\) and \(\|x+x_n\|\rightarrow 2\), where \(S_X\) denotes the unit sphere of \(X\). If the convergence of the subsequence is in the weak topology, the space is said to be WCLR (a concept due to L. P. Vlasov). In the present paper, the class of weakly CLUR is introduced: As above, the sequence \(\{x_n\}\) has a subsequence that weakly converges, but now the weak limit is in \(A_0(x):=\bigcup_{x^*\in J(x)} S(X,x^*,0)\), where \(J(x):=\{x^*\in S_{X^*}:\;x^*(x)=1\}\), and \(S(X,x^*,\delta):=\{y\in B_{X}:\;x^*(y)\geq 1-\delta\}\) for \(\delta\geq 0\). Two examples are presented to help separating some of those concepts: Both are renormings of \(\ell_2\). The first one presents a (reflexive) space with the Kadets-Klee property (KK, in short, i.e., on \(S_X\) the weak and norm-convergent sequences coincide) that is not weakly CLUR. The second one is (reflexive and) weakly CLUR, although it is neither CLUR nor WLUR. The remainder of the paper deals with characterizations of the weakly CLUR and the CLUR properties, gives a sufficient condition for weakly CLUR on reflexive spaces in terms of the KK property, and, alternatively, in terms of the local nearly uniformly smooth property or in terms of the strong subdifferentiability of the norm together with the compactness of the duality mapping (two sets of conditions that, according to a later proposition, turn out to be the same).
weakly compactly locally uniformly convex
scientific article; zbMATH DE number 6813447
scientific article