. "IAA100190801" . . "C\u00EDlem projektu je studovat mo\u017En\u00E1 zobecn\u011Bn\u00ED klasick\u00FDch v\u011Bt kone\u010Dn\u011B dimenzion\u00E1ln\u00ED anal\u00FDzy na p\u0159\u00EDpad funkc\u00ED na Banachov\u00FDch prostorech. Jde p\u0159edev\u0161\u00EDm o ot\u00E1zky aproximac\u00ED pomoc\u00ED v\u00EDcen\u00E1sobn\u011B hladk\u00FDch funkc\u00ED, typu Stone Weierstrassovy v\u011Bty, a s t\u00EDm \u00FAzce souvisej\u00EDc\u00ED problematikou vlastnost\u00ED polynom\u016F na Banachov\u00FDch prostorech. Jako konkr\u00E9tn\u00ED p\u0159\u00EDklad lze uv\u00E9st ot\u00E1zky n\u00E1sleduj\u00EDc\u00EDho typu: pokud na Banachov\u011B prostoru existuje separuj\u00EDc\u00ED polynom, existuje t\u00E9\u017E konvexn\u00ED separuj\u00EDc\u00ED polynom, a obecn\u011Bji vypl\u00FDv\u00E1 z existence hladk\u00FDch bump\u016F t\u00E9\u017E hladk\u00E1 renormace? Za jak\u00FDch podm\u00EDnek lze aproximovat funkce spolu s derivacemi vy\u0161\u0161\u00EDho \u0159\u00E1du? Plat\u00ED na Hilbertov\u011B prostoru verze Alexandrovovy v\u011Bty? Lze na prostoru c_0 aproximovat pomoc\u00ED re\u00E1ln\u011B analytick\u00FDch funkc\u00ED, a existuj\u00ED zde velmi hladk\u00E9 body pro ka\u017Edou konvexn\u00ED funkci? Existuje charakterizace polyhedr\u00E1ln\u00EDch Orliczov\u00FDch prostor\u016F pomoc\u00ED Orliczovsk\u00E9 funkce? V\u011Bt\u0161ina t\u011Bchto probl\u00E9m\u016F je v\u0161eobecn\u011B zn\u00E1ma specialist\u016Fm v oboru, a objevuje se opakovan\u011B v literatu\u0159e." . . "Byl nalezen siln\u00FD protip\u0159\u00EDklad na Peanovu v\u011Btu v nekone\u010Dn\u00E9 dimenzi. Byla vybudov\u00E1na teorie omega limitn\u00EDch mno\u017Ein v nekone\u010Dn\u00E9 dimenzi, teorie aproximac\u00ED lipschitzovsk\u00FDch oper\u00E1tor\u016F , bylo vy\u0159e\u0161eno n\u011Bkolik probl\u00E9m\u016F z renormac\u00ED a teorie oper\u00E1tor\u016F."@cs . "A strong counterexample to Peano theorem was found in infinite dimension. A theory of omega limit sets, and of approximation of Lipschitz operators were created. Several problems on renormings and operator theory were solved."@en . " polynomial" . "11"^^ . . "11"^^ . . "smoothness" . "2013-06-28+02:00"^^ . "2"^^ . . "smoothness; polynomial; approximation; point of differentiability"@en . "0"^^ . . " approximation" . "2010-03-09+01:00"^^ . "2010-12-31+01:00"^^ . "Hladkost v Banachov\u00FDch prostorech" . . . "http://www.isvav.cz/projectDetail.do?rowId=IAA100190801"^^ . "1"^^ . "We intend to study the possible generalizations of the classical theorems of finite dimensional analysis to the setting of Banach spaces. We are mostly concerned with smooth approximations, in the spirit of the Stone Weierstrass theorem, and the closely connected study of polynomials on Banach spaces. Let us state a few typical problems. Does the existence of a separating polynomial on a Banach space imply the existence of a convex and separating polynomial, or more generaly is there a way to obtain convex higher smooth functions from higher smooth bumps? When are uniform approximations of a function together with its higher derivatives possible? Does Alexandroff theorem hold on a Hilbert space? Are real analytic approximations possible on c_0, and are there very smooth points for convex function threon? Is there a characterization of polyhedrality for Orlicz spaces using the Orlicz function? Most of these problems are well known to the specialists and permeate the literature."@en . . . "Smoothness in Banach spaces"@en . . . . . "2008-01-01+01:00"^^ . . . .