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 community uniformly rotund norms on Banach areas. The publication bargains new ideas for renorming difficulties, them all according 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 neighborhood uniformly rotund renormings on X, are studied and a brand new body for the idea is acquired, with interaction among useful research, optimization and topology utilizing subdifferentials of Lipschitz services and masking equipment of metrization concept. 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 will be faraway from linear, for example the duality map from X to X* offers a non-linear instance whilst the norm in X is Fréchet differentiable.

This quantity might be attention-grabbing for the wide spectrum of experts operating in Banach area concept, and for researchers in limitless dimensional practical analysis.

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

**Example text**

Let us suppose that for any pair s, t ∈ K for which there exists γ ∈ Λ with s(γ) = t(γ) we can ﬁnd x ∈ A with x(s) = x(t). Then A is uniformly dense in {y ◦ (PΛ K ) : y ∈ C (PΛ K)}. In particular, if Λn ⊂ Γ and Λ = {y ◦ (PΛn K) is uniformly dense in {y ◦ (PΛ ∞ n=1 Λn then the algebra generated by : y ∈ C (PΛn K) , n ∈ N} K) : y ∈ C (PΛ K)}. Proof. 18) it follows that there exists an algebra B, B ⊂ C (PΛ K), which contains the constant functions such that A = {y ◦ (PΛ K ) : y ∈ B} . Since B separates points in PΛ K the Stone-Weierstrass theorem gives us that B is uniformly dense in C (PΛ K) and the ﬁrst part of the statement is proved.

30 2 σ-Continuous and Co-σ-continuous Maps Proof. 40. 43. Let A be a subset of a locally convex linear topological space and let H be the family of all open half spaces in X. If (Y, ) is a metric space and Φ : A → Y is a σ-continuous map then Φ is σ-slicely continuous if there exists a sequence {An : n ∈ N} of subsets of A such that the family {An ∩ H : H ∈ H, n ∈ N} is a network for the topology of A. Proof. 39. A set of a linear topological space X is said to be radial (with respect to 0) if for every x ∈ X there exists λ > 0 such that λx ∈ A.

29) we have |x (t0 ) − x (s0 )| < ε . 30) |x(t) − x(s)| ≤ |x(t) − x(t0 )| + |x(t0 ) − x(s0 )| + |x(s0 ) − x(s)| < ε ε + . 2 2 This proves the claim. 71 it is enough to repeat the argument of the above claim. 1 Discreteness and Network Conditions A class of generalized metric spaces is a class of spaces deﬁned by a property shared by all metric spaces which is close to metrizability in some sense [Gru84]. e. a topological space is a σ-space if it has a σ-discrete network. Here we shall deal with a further reﬁnement replacing discrete by isolated or slicely isolated.