By Aníbal Moltó, José Orihuela, Stanimir Troyanski, Manuel Valdivia

Abstract topological instruments from generalized metric areas are utilized during this quantity to the development of in the neighborhood uniformly rotund norms on Banach areas. The ebook deals new strategies for renorming difficulties, them all in keeping with a community research for the topologies concerned contained in the problem.

Maps from a normed area X to a metric house Y, which offer in the neighborhood uniformly rotund renormings on X, are studied and a brand new body for the speculation is got, with interaction among practical research, optimization and topology utilizing subdifferentials of Lipschitz capabilities and overlaying tools of metrization idea. Any one-to-one operator T from a reflexive area X into c_{0} (T) satisfies the authors' stipulations, shifting the norm to X. however the authors' maps should be faraway from linear, for example the duality map from X to X* supplies a non-linear instance whilst the norm in X is Fréchet differentiable.

This quantity might be attention-grabbing for the vast spectrum of experts operating in Banach area conception, and for researchers in limitless dimensional sensible analysis.

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Negrepontis [MN92] and the paper of V. Zizler in the Handbook of the Geometry of Banach spaces vol. II [Ziz03], as well as the monographs by R. Deville, G. Godefroy and V. Zizler [DGZ93] and by M. Fabian [Fab97]. Several deﬁnitions can be given for this notion (see [Fab97, Chap. 57. A topological space (X, T ) is called K-countably determined if it embeds in a compact space K in such a way that there is a sequence Kn of compact subsets of K such that for every x ∈ X and y ∈ K \ X there exists n0 ∈ N such that x ∈ Kn0 and y ∈ / Kn0 .

4) we get #Vx ≤ ℵ0 . 9) #Zx ≤ ℵ0 . Let x, xn ∈ C0 (Υ) and T xn − T x ∞ → 0. Then supp T x ⊂ T xn . From the deﬁnition of Zx we have ∞ m=1 ∞ n=m supp ∞ Zx ⊂ Zxn . 33 as soon as we show that · ∞ . 11) x ∈ spanZx Fix x ∈ C0 (Υ). For ε > 0 set Lε (|x|) = {t ∈ Υ : |x(t)| ≥ ε}, we show that there exists Dx,ε ⊂ Υ such that #Dx,ε < ∞ , Dx,ε ⊂ supp T x , Lε (|x|) ⊂ (0, v] . 12) 38 2 σ-Continuous and Co-σ-continuous Maps Since Lε (|x|) is compact and {(0, v] : v ∈ Lε (|x|)} is an open cover of Lε (|x|) k we can ﬁnd u1 , u2 , .

X = From the choice of the sets Zx ’s we must have x ∈ Zx so Λx controls x. Moreover let xn ∈ C(K) be a sequence such that Φxn → Φx then x∈ Then ∞ n=1 {Zxn : n ∈ N} = {Mxn : n ∈ N} . Λxn controls x. ii)⇒i). For x ∈ C(K) let Zx := {y ◦ (PΛx K ) : y ∈ C (PΛx K)}. Since PΛx K ⊂ [0, 1]Λx and Λx is countable we have that the space PΛx K is metric so Zx is separable. 33 let us consider a sequence x, xn ∈ C(K) such that lim Φxn = Φx. From ii) it follows that Λ := Λxn controls x so there must exist y ∈ C(PΛ K) such that x = y ◦ (PΛ K ).