About: Geometry and Gâteaux smoothness in separable Banach spaces

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rdf:type	skos:Concept http://linked.opendata.cz/ontology/domain/vavai/Vysledek
Description	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. 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. (en)
Title	Geometry and Gâteaux smoothness in separable Banach spaces Geometry and Gâteaux smoothness in separable Banach spaces (en)
skos:prefLabel	Geometry and Gâteaux smoothness in separable Banach spaces Geometry and Gâteaux smoothness in separable Banach spaces (en)
skos:notation	RIV/67985840:_____/12:00378924!RIV13-AV0-67985840
http://linked.open...avai/predkladatel	Matematický ústav AV ČR, v. v. i.
http://linked.open...avai/riv/aktivita	P Z
http://linked.open...avai/riv/aktivity	P(GAP201/11/0345), P(IAA100190901), Z(AV0Z10190503)
http://linked.open...iv/cisloPeriodika	2
http://linked.open...vai/riv/dodaniDat	2013
http://linked.open...aciTvurceVysledku	Zizler, Václav Hájek, Petr
http://linked.open.../riv/druhVysledku	J - Článek v odborném periodiku
http://linked.open...iv/duvernostUdaju	S - Úplné a pravdivé údaje nepodléhající ochraně podle zvláštních právních předpisů
http://linked.open...titaPredkladatele	Matematický ústav AV ČR, v. v. i.
http://linked.open...dnocenehoVysledku	138085
http://linked.open...ai/riv/idVysledku	RIV/67985840:_____/12:00378924
http://linked.open...riv/jazykVysledku	eng - angličtina
http://linked.open.../riv/klicovaSlova	Gâteaux differentiable norms; extreme points; Radon-Nikodým property (en)
http://linked.open.../riv/klicoveSlovo	Radon-Nikodým property extreme points Gâteaux differentiable norms
http://linked.open...odStatuVydavatele	HR - Chorvatská republika
http://linked.open...ontrolniKodProRIV	[1AD61D1B19A8]
http://linked.open...i/riv/nazevZdroje	Operators and Matrices
http://linked.open...in/vavai/riv/obor	BA
http://linked.open...ichTvurcuVysledku	2 (xsd:int)
http://linked.open...cetTvurcuVysledku	3 (xsd:int)
http://linked.open...vavai/riv/projekt	Topological and geometric structures in Banach spaces Nonlinear functional analysis
http://linked.open...UplatneniVysledku	2012
http://linked.open...v/svazekPeriodika	6
http://linked.open...iv/tvurceVysledku	Montesinos, V. Zizler, Václav Hájek, Petr Pavel
http://linked.open...ain/vavai/riv/wos	000306349600001
http://linked.open...n/vavai/riv/zamer	Rozvoj a prohloubení obecných matematických poznatků a jejich užití v dalších vědních oborech a v praxi
issn	1846-3886
number of pages	32 (xsd:int)
http://bibframe.org/vocab/doi	10.7153/oam-06-15
is http://linked.open...avai/riv/vysledek of	Geometry and Gâteaux smoothness in separable Banach spaces

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