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

ISBN-10: 3540850309

ISBN-13: 9783540850304

ISBN-10: 3540850317

ISBN-13: 9783540850311

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 publication deals new recommendations for renorming difficulties, them all in response to a community research for the topologies concerned contained in the problem.

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

This quantity should be fascinating for the large spectrum of experts operating in Banach house thought, and for researchers in countless dimensional useful analysis.

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**Read e-book online A Nonlinear Transfer Technique for Renorming PDF**

Summary 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 booklet deals new strategies for renorming difficulties, them all in line with a community research for the topologies concerned contained in the challenge. Maps from a normed area X to a metric area Y, which offer in the neighborhood uniformly rotund renormings on X, are studied and a brand new body for the speculation is received, with interaction among useful research, optimization and topology utilizing subdifferentials of Lipschitz services and overlaying equipment of metrization conception.

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**Additional info for A Nonlinear Transfer Technique for Renorming**

**Example text**

68 there are compacta K for which C(K) admits a LUR renorming and we can ﬁnd some x ∈ C(K), x = 0, such that supp Ωx = ∅ so supp Ωx does not control x. 67 is essential to get a LUR renorming. g. [Kel55, p 164]. Then for H we can take [0, 1] as the set Γ so it makes sense to write Ωx(γ) for γ ∈ [0, 1]. 70. Let x : H → R be the real function deﬁned by the formula 1 t(γ)dγ . x(t) = 0 It is obvious that x ∈ C(H) and Ωx(γ) = 0 for every γ ∈ [0, 1]. Therefore supp Ωx does not control x. 7 Co-σ-continuous Maps in C(K) 45 The next lemma illustrates how the set supp Ωx can be enlarged to control x.

Proof. If Φ : ΨY → Y is a selector and we take Z := Φ(ΨY ) ⊂ Y then Ψ Z is one-to-one and co-σ-continuous and the inverse map, which coincides with Φ, must be σ-continuous. 30. Let (Y, ) be a metric space and let (X, T ) be a topological space. For a map Ψ : Y → X the following assertions are equivalent: i) Ψ is co-σ-continuous. ii) The ﬁbers of Ψ are separable and every selector Φ of Ψ−1 is σ-continuous. Proof. 29. ii)=⇒i) Let us choose for every y ∈ Y a sequence {yn : n ∈ N} which is dense in the ﬁber Ψ−1 (Ψy).

I)⇒ii). According to the theorem of Mibu for any x ∈ C(K) we can and do select a countable subset ∆x ⊂ Γ which controls x. 32. For each x let Mx a ﬁxed countable dense subset of Zx . Set {∆y : y ∈ Mx } . Λx = 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.

### A Nonlinear Transfer Technique for Renorming by Aníbal Moltó, José Orihuela, Stanimir Troyanski, Manuel Valdivia

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