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

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 e-book bargains new innovations 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 house Y, which offer in the community uniformly rotund renormings on X, are studied and a brand new body for the idea is got, with interaction among useful research, optimization and topology utilizing subdifferentials of Lipschitz features and overlaying equipment of metrization idea. Any one-to-one operator T from a reflexive house X into c_{0} (T) satisfies the authors' stipulations, moving 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 while the norm in X is Fréchet differentiable.

This quantity can be fascinating for the huge spectrum of experts operating in Banach house thought, and for researchers in countless dimensional sensible analysis.

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**Extra resources for A Nonlinear Transfer Technique for Renorming**

**Example text**

61 then Λ controls x if, and only if, for every ε > 0 there exist δ > 0 and F ⊂ Λ, F ﬁnite, such that |x(s) − x(t)| < ε whenever |s(γ) − t(γ)| < δ for every γ ∈ F . Proof. It is clear that any set Λ which fulﬁlls the last condition must control x. Conversely if Λ controls x let us assume there exists ε0 > 0 such that for any ﬁnite subset F ⊂ Λ and any n ∈ N we can ﬁnd sn,F , tn,F ∈ K with |x (sn,F ) − x (tn,F )| ≥ ε0 and |sn,F (γ) − tn,F (γ)| < 1/n for any γ ∈ F . The compactness of K gives cluster points s and t to the nets (sn,F ) and (tn,F ).

For every positive rational number q the map Ψq x := qΨ1 x is also σ-slicely continuous in X \ {0}. Finally since Ψ is homogeneous we have Ψx = lim q→1/r(x) Ψq x for every x ∈ X \ {0} · consequently Ψx ∈ {Ψq x : q ∈ Q+ } for every x ∈ X \ {0}. 23 shows that Ψ X\{0} is σ-slicely continuous and so is Ψ. 3 for the cluster point argument to ﬁnish the proof. 6 that the only assumption we need on a linear continuous map T from a normed space X into a LUR renormable space Y to lift the LUR norm on X is the co-σ-continuity of T .

N→∞ Proof. 30. 13 to deduce the existence of a countable subset WΨy that Φn (Ψy) ∈ n : p∈N Φn WΨy p n whenever limp→∞ Ψyp = Ψy. If we set Zy := Ψ−1 (Ψy)∪ Φn WΨy : n∈N for y ∈ Y then Zy is a separable subset of Y . 30 y ∈ {Φn Ψy : n ∈ N} ⊂ Zyp : p ∈ N . 2 Co-σ-continuous Maps 27 ii)=⇒i) It is clear that condition ii) implies Ψ−1 (Ψy) ⊂ Zy for every y ∈ Y so the ﬁbers of Ψ must be separable. Moreover if for every y ∈ Y we select a countable subsets Wy ⊂ Zy dense in Zy then for every selector Φ of Ψ−1 we have {Wyk : k ∈ N} Φ(Ψy) ∈ whenever limk→∞ Ψyk = Ψy.