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 booklet deals new innovations for renorming difficulties, them all in accordance with 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 bought, with interaction among practical research, optimization and topology utilizing subdifferentials of Lipschitz capabilities and overlaying equipment of metrization thought. 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 will be faraway from linear, for example the duality map from X to X* provides a non-linear instance whilst the norm in X is Fréchet differentiable.

This quantity could be fascinating for the large spectrum of experts operating in Banach house idea, and for researchers in endless dimensional practical analysis.

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

**Example text**

7) and the maximality of v0 we get Fv0 ∩ supp x = ∅. 8) we get x (v0 ) = 0. 56. 5) is co-σ-continuous. Proof. 33 let us deﬁne for every x ∈ C0 (Υ) the sets Ux = 1l(0,u] : ∃v, w ∈ supp T x, (0, u] = (0, v] ∩ (0, w] Vx = 1l(0,u] : ∃v ∈ supp T x, u ∈ (0, v], Fu ∩ (0, v] = ∅ Zx = Ux ∪ Vx . Since T x ∈ c0 (Υ) we have #supp T x ≤ ℵ0 . Hence #Ux ≤ ℵ0 . 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 .

V) Every discrete family of subsets of X for the norm topology is σ-slicely isolatedly decomposable in σ(X, F ). ∞ vi) The norm topology of X admits a network N = n=1 Nn where every Nn veriﬁes σ(X,F ) A ∩ conv ({B : B ∈ Nn , B = A}) = ∅ for every A ∈ Nn . A. , A Nonlinear Transfer Technique for Renorming. Lecture Notes in Mathematics 1951, c Springer-Verlag Berlin Heidelberg 2009 49 50 3 Generalized Metric Spaces and Locally Uniformly Rotund Renormings vii)There is a sequence {An : n ∈ N} of subsets of X such that the family {An ∩ H : H is a σ(X, F )-open half space, n ∈ N} is a network for the norm topology viii)The identity map Id : (X, σ(X, F )) → (X, · ) is σ-continuous and there is a sequence {An : n ∈ N} of subsets of X such that the family {An ∩ H : H is a σ(X, F )-open half space, n ∈ N} is a network for the σ(X, F )-topology Proof.

If this is not the case there must exist a net {xβ }β with 52 3 Generalized Metric Spaces and Locally Uniformly Rotund Renormings |xβ | ≤ 1 = |x0 | and g (xβ ) ≤ µ for every β such that σ(X, F ) − limβ xβ = x0 . Since | · | is σ(X, F )-lower semicontinuous we have 1 = |x0 | ≥ lim sup |xβ | ≥ lim inf |xβ | ≥ |x0 | β β so limβ |xβ | = |x0 |. Our assumption on SX gives that weak − lim xβ = x0 β which contradicts that x0 ∈ L and xβ ∈ X \ L for all β. The theorem of HahnBanach in the duality < X, F > tells us that there exists f ∈ F separating σ(X,F ) BX \ L from x0 so x0 ∈ {x ∈ X : f (x) > ξ} ∩ BX ⊂ L.