Differentiable Functions and Rough Norms on Banach Spaces

We investigate sufficient and necessary conditions for the space of bounded linear operators between two Banach spaces to be rough or average rough. Our main result is that L(X,Y) is δ‐average rough whenever X* is δ‐average rough and Y is alternatively octahedral. This allows us to give a unified improvement of two theorems by Becerra Guerrero, López‐Pérez, and Rueda Zoca [J. Math. Anal. Appl. 427 (2015)].

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“…Definition 1.1 Let X be a Banach space and δ > 0. The space X is said to be r δ-rough [7] if, for every x ∈ S X , lim sup y →0…”

Section: Introductionmentioning

confidence: 99%

Rough norms in spaces of operators

Haller

Langemets

Põldvere

2017

Mathematische Nachrichten

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“…If X is separable but X Ã is not separable, then there is a Lipschitz (even convex) function from X to R which fails to have a point of 1-Fre Âchet differentiability. This theorem is proved in [15] (see also Proposition 4.12 in [2] and the remark following its proof ).…”

Section: Introductionmentioning

confidence: 84%

Almost Fréchet Differentiability of Lipschitz Mappings Between Infinite-Dimensional Banach Spaces

Johnson¹,

Lindenstrauss²,

Preiss³

et al. 2002

Proc. Lond. Math. Soc.

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We give several sufficient conditions on a pair of Banach spaces X and Y under which each Lipschitz mapping from a domain in X to Y has, for every $\epsilon > 0$, a point of $\epsilon$-Fréchet differentiability. Most of these conditions are stated in terms of the moduli of asymptotic smoothness and convexity, notions which have appeared in the literature under a variety of names. We prove, for example, that for $\infty > r > p \ge 1$, every Lipschitz mapping from a domain in an $\ell_r$-sum of finite-dimensional spaces into an $\ell_p$-sum of finite-dimensional spaces has, for every $\epsilon > 0$, a point of $\epsilon$-Fréchet differentiability, and that every Lipschitz mapping from an asymptotically uniformly smooth space to a finite-dimensional space has such points. The latter result improves, with a simpler proof, an earlier result of the second and third authors. We also survey some of the known results on the notions of asymptotic smoothness and convexity, prove some new properties, and present some new proofs of existing results.2000 Mathematical Subject Classification: 46G05, 46T20.

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“…It is not difficult to see that the arguments of Lemma 1 and Theorem 1 may be repeated for higher ordinals. In particular, this argument will give a proof of the complex version of a theorem proved by Leach and Whitfield [8] in the real case: Theorem 4. Let X be a separable Banach space and Y a Banach space and T: X -*■ Y such that T*(Y*) is nonseparable.…”

Section: Corollarymentioning

confidence: 94%