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 e-book 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 supply in the neighborhood uniformly rotund renormings on X, are studied and a brand new body for the speculation is got, with interaction among useful research, optimization and topology utilizing subdifferentials of Lipschitz capabilities and masking equipment of metrization idea. 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 might 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 could be attention-grabbing for the vast spectrum of experts operating in Banach house idea, and for researchers in limitless dimensional practical analysis.

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

**Example text**

A tree is a partially ordered set (Υ, ) with the property that, for every t ∈ Υ, the set {s ∈ Υ : s t} is well-ordered by . It is used normal interval notation, so that, for instance, (s, u] = {t ∈ Υ : s ≺ t u}. For convenience of notation, the tree Υ has two “imaginary” elements, not in Υ, denoted by 0 and ∞, and having the property that 0 ≺ t ≺ ∞ for all t ∈ Υ. This allows us to extend our interval notation to include expressions like (0, t] and [t, ∞). Note that, by deﬁnition, each (0, t] is well-ordered, but that [t, ∞) need not be.

For instance we have the following result of M. Raja [Raj02]. 7. If X is a Banach space such that the weak∗ and the weak topologies coincide on the dual unit sphere then X ∗ has an equivalent dual LUR norm. Proof. By [NP75] X ∗ has the Radon-Nikodym property and by the result of M. Fabian and G. Godefroy [FG88] X ∗ has a LUR equivalent norm. 8. A function Φ from a locally convex linear topological space X into a metric space (Y, ) is said to be ε-σ-slicely continuous for a ﬁxed ∞ ε > 0 if we can decompose X = n=1 Xn in such a way that for every n ∈ N and every x ∈ Xn there is an open half space H in X with x ∈ H and such that osc (Φ H∩Xn ) < ε.

Therefore the sequence of sets Aq1 ∩ Aq2 ∩ . . ∩ Aqn : qi ∈ Q+ , n ∈ N gives us the property P (weak topology , weak topology) for the map Φ. 46. Let X be a locally convex linear topological space and let A ⊂ X be a radial subset of X. If Y is a normed space and Ψ : X → Y is a homogeneous map then Ψ is σ-slicely continuous (respectively σ-continuous) if, and only if, its restriction to A is σ-slicely continuous (respectively σ-continuous ). Proof. 44 has P (H, H). 40. For every positive rational number q the map Ψq x := qΨ1 x is also σ-slicely continuous in X \ {0}.

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

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