About: Gateaux and Hadamard Differentiability via Directional Differentiability

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rdf:type	skos:Concept http://linked.opendata.cz/ontology/domain/vavai/Vysledek
Description	Let X be a separable Banach space, Y a Banach space and f : X -> Y an arbitrary mapping. Then the following implication holds at each point x is an element of X except a sigma-directionally porous set: If the one-sided Hadamard directional derivative f(H+)'(x,u) exists in all directions u from a set S-x subset of X whose linear span is dense in X, then f is Hadamard differentiable at x. This theorem improves and generalizes a recent result of A. D. Ioffe, in which the linear span of S-x equals X and Y = R. An analogous theorem, in which f is pointwise Lipschitz, and which deals with the usual one-sided derivatives and Gateaux differentiability is also proved. It generalizes a result of D. Preiss and the author, in which f is supposed to be Lipschitz. Let X be a separable Banach space, Y a Banach space and f : X -> Y an arbitrary mapping. Then the following implication holds at each point x is an element of X except a sigma-directionally porous set: If the one-sided Hadamard directional derivative f(H+)'(x,u) exists in all directions u from a set S-x subset of X whose linear span is dense in X, then f is Hadamard differentiable at x. This theorem improves and generalizes a recent result of A. D. Ioffe, in which the linear span of S-x equals X and Y = R. An analogous theorem, in which f is pointwise Lipschitz, and which deals with the usual one-sided derivatives and Gateaux differentiability is also proved. It generalizes a result of D. Preiss and the author, in which f is supposed to be Lipschitz. (en)
Title	Gateaux and Hadamard Differentiability via Directional Differentiability Gateaux and Hadamard Differentiability via Directional Differentiability (en)
skos:prefLabel	Gateaux and Hadamard Differentiability via Directional Differentiability Gateaux and Hadamard Differentiability via Directional Differentiability (en)
skos:notation	RIV/00216208:11320/14:10285323!RIV15-MSM-11320___
http://linked.open...avai/riv/aktivita	I P
http://linked.open...avai/riv/aktivity	I, P(GAP201/12/0436)
http://linked.open...iv/cisloPeriodika	3
http://linked.open...vai/riv/dodaniDat	2015
http://linked.open...aciTvurceVysledku	Zajíček, Luděk
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	Univerzita Karlova v Praze / Matematicko-fyzikální fakulta
http://linked.open...dnocenehoVysledku	17957
http://linked.open...ai/riv/idVysledku	RIV/00216208:11320/14:10285323
http://linked.open...riv/jazykVysledku	eng - angličtina
http://linked.open.../riv/klicovaSlova	pointwise Lipschitz mapping; sigma-directionally porous set; Hadamard directional derivatives; directional derivatives; Hadamard differentiability; Gateaux differentiability (en)
http://linked.open.../riv/klicoveSlovo	Hadamard differentiability Hadamard directional derivatives pointwise Lipschitz mapping sigma-directionally porous set Gateaux differentiability directional derivatives
http://linked.open...odStatuVydavatele	DE - Spolková republika Německo
http://linked.open...ontrolniKodProRIV	[A65D7B9D8525]
http://linked.open...i/riv/nazevZdroje	Journal of Convex Analysis
http://linked.open...in/vavai/riv/obor	BA
http://linked.open...ichTvurcuVysledku	1 (xsd:int)
http://linked.open...cetTvurcuVysledku	1 (xsd:int)
http://linked.open...vavai/riv/projekt	Teorie reálných funkcí a deskriptivní teorie množin III
http://linked.open...UplatneniVysledku	2014
http://linked.open...v/svazekPeriodika	21
http://linked.open...iv/tvurceVysledku	Zajíček, Luděk
http://linked.open...ain/vavai/riv/wos	000342730400006
issn	0944-6532
number of pages	11 (xsd:int)
http://localhost/t...ganizacniJednotka	11320

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