Points of differentiability of the norm in Lipschitz-free spaces

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:489

Article

Aliaga, Ramon J.¹  Rueda Zoca, Abraham² 

[1] Univ Politecn Valencia, Inst Univ Matemat Pura & Aplicada, Camino Vera S-N, Valencia 46022, Spain

[2] Univ Granada, Fac Ciencias, Dept Anal Matemat, Granada 18071, Spain

关键词: Frechet differentiability;    Gateaux differentiability;    Lipschitz-free space;   

DOI  :  10.1016/j.jmaa.2020.124171

来源: Elsevier

【 摘 要 】

We consider convex series of molecules in Lipschitz-free spaces, i.e. elements of the form mu = Sigma(n) lambda(n) delta(xn)- delta(yn/)d(x(n),y(n)) such that vertical bar vertical bar mu vertical bar vertical bar = Sigma(n) vertical bar lambda(n)vertical bar. We characterise these elements in terms of geometric conditions on the points xn, ynof the underlying metric space, and determine when they are points of Gateaux differentiability of the norm. In particular, we show that Gateaux and Frechet differentiability are equivalent for finitely supported elements of Lipschitz-free spaces over uniformly discrete and bounded metric spaces, and that their tensor products with Gateaux (resp. Frechet) differentiable elements of a Banach space are Gateaux (resp. Frechet) differentiable in the corresponding projective tensor product. (C) 2020 Elsevier Inc. All rights reserved.

【 授权许可】

Free