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Grantor: Grant Agency of Czech Academy of Sciences

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.

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