Geometry and Gâteaux smoothness in separable Banach spaces

0378924 - MÚ 2013 RIV HR eng J - Journal Article
Hájek, Petr Pavel - Montesinos, V. - Zizler, Václav
Geometry and Gâteaux smoothness in separable Banach spaces.
Operators and Matrices. Roč. 6, č. 2 (2012), s. 201-232. ISSN 1846-3886. E-ISSN 1846-3886
R&D Projects: GA ČR(CZ) GAP201/11/0345; GA AV ČR IAA100190901
Institutional research plan: CEZ:AV0Z10190503
Keywords : Gâteaux differentiable norms * extreme points * Radon-Nikodým property
Subject RIV: BA - General Mathematics
Impact factor: 0.529, year: 2012
http://oam.ele-math.com/06-15/Geometry-and-Gateaux-smoothness-in-separable-Banach-spaces

It is a classical fact, due to Day, that every separable Banach space admits an equivalent Gateaux smooth renorming. In fact, it admits an equivalent uniformly Gateaux smooth norm, as was shown later by Day, James, Swaminathan, and independently by the third named author. It is therefore rather unexpected that the existence of Gateaux smooth renormings satisfying various quantitative estimates on the directional derivative has rather strong structural and geometrical implications for the space. For example, by a result of Vanderwerff, if the directional derivatives satisfy a p-estimate, where p varies arbitrarily with respect to the point and the direction in question, then the Banach space must be an Asplund space. In the present survey paper, we discuss the interplay between various types of Gateaux differentiability of norms and extreme points with the geometry of separable Banach spaces.
Permanent Link: http://hdl.handle.net/11104/0210236