0025746 - MÚ 2006 RIV IL eng J - Journal Article
Deville, R. - Hájek, Petr Pavel
On the range of the derivative of Gâteaux-Smooth functions on separable Banach spaces.
[Obraz derivace Gateauovsky hladkých funkcí na separabilních Banachových prostorech.]
Israel Journal of Mathematics. Roč. 145, č. 2 (2005), s. 257-269. ISSN 0021-2172. E-ISSN 1565-8511
R&D Projects: GA AV ČR(CZ) IAA1019003; GA AV ČR(CZ) IAA1019205; GA ČR(CZ) GA201/01/1198
Institutional research plan: CEZ:AV0Z10190503
Keywords : Gâteaux-Smooth functions * Banach space * Lipschitz function
Subject RIV: BA - General Mathematics
Impact factor: 0.448, year: 2005

We prove that there exists a Lipschitz function from l1 into R2 which is Gâteaux-differentiable at every point and such that for every x,y .. l1, the norm of f´(x) - f´(y) is bigger than 1. On the other hand, for every Lipschitz and Gâteaux-differentiable function from an arbitrary Banach space X into R and for every .. > 0, there always exist two points x,y .. X such that ¦¦f´(x)-f´(y)¦¦ is less than .. . We also construct, in every infinite dimensional separable Banach space, a real valued function f on X, which is Gâteaux-differentiable at every point, has bounded non-empty support, and with the properties that f´is norm to weak* continuous and f´(X) has an isolated point a, and that necessarily a .. 0.

Hlavní výsledek je příklad Lipschitzovské a Gâteauovsky hladké funkce z 1_1 do R^2 takové, že derivace v libovolných dvou různých bodech mají od sebe vzdálenost aspoň jedna.
Permanent Link: http://hdl.handle.net/11104/0116106