A Characterization of C -Smooth Banach Spaces (original) (raw)

On nicely smooth Banach spaces

1996

In this work, we obtain some necessary and some sufficient conditions for a space to be nicely smooth, and show that they are equivalent for separable or Asplund spaces. We obtain a sufficient condition for the Ball Generated Property (BGP), and conclude that Property (II)(II)(II) implies the BGP, which, in turn, implies the space is nicely smooth. We show that

Smoothness in Banach spaces. Selected problems

2006

This is a short survey on some recent as well as classical results and open problems in smoothness and renormings of Banach spaces. Applications in general topology and nonlinear analysis are considered. A few new results and new proofs are included. An effort has been made that a young researcher may enjoy going through it without any special pre-requisites and

On the range of the derivatives of a smooth function between Banach spaces

Mathematical Proceedings of the Cambridge Philosophical Society, 2003

We study the size of the range of the derivatives of a smooth function between Banach spaces. We establish conditions on a pair of Banach spaces X and Y to ensure the existence of a C p smooth (Fréchet smooth or a continuous Gâteaux smooth) function f from X onto Y such that f vanishes outside a bounded set and all the derivatives of f are surjections. In particular we deduce the following results. For the Gâteaux case, when X and Y are separable and X is infinite-dimensional, there exists a continuous Gâteaux smooth function f from X to Y , with bounded support, so that f (X) = L(X, Y). In the Fréchet case, we get that if a Banach space X has a Fréchet smooth bump and dens X = dens L(X, Y), then there is a Fréchet smooth function f : X −→ Y with bounded support so that f (X) = L(X, Y). Moreover, we see that if X has a C p smooth bump with bounded derivatives and dens X = dens L m s (X; Y) then there exists another C p smooth function f : X −→ Y so that f (k) (X) = L k s (X; Y) for all k = 0, 1, ..., m. As an application, we show that every bounded starlike body on a separable Banach space X with a (Fréchet or Gâteaux) smooth bump can be uniformly approximated by smooth bounded starlike bodies whose cones of tangent hyperplanes fill the dual space X *. In the non-separable case, we prove that X has such property if X has smooth partitions of unity.

Smooth bump functions and geomentry of Banach spaces

Mathematika, 1993

Norms with moduli of smoothness of power type are constructed on spaces with the Radon-Nikodym property that admit pointwise Lipschitz bump functions with pointwise moduli of smoothness of power type. It is shown that no norms with pointwise moduli of rotundity of power type can exist on nonsuperreflexive spaces. A new smoothness characterization of spaces isomorphic to Hilbert spaces is given.

Pyramidal vectors and smooth functions on Banach spaces

Proceedings of the American Mathematical Society, 2000

We prove that if X X , Y Y are Banach spaces such that Y Y has nontrivial cotype and X X has trivial cotype, then smooth functions from X X into Y Y have a kind of “harmonic" behaviour. More precisely, we show that if Ω \,\Omega is a bounded open subset of X X and f : Ω ¯ → Y f:{\overline {\Omega }}\to Y is C 1 C^{1} - \, smooth with uniformly continuous Fréchet derivative, then f ( ∂ Ω ) f(\partial \Omega ) is dense in f ( Ω ¯ ) f({\overline {\Omega }}) . We also give a short proof of a recent result of P. Hájek.

On the range of the derivatives of a smooth function betweenBanach spacesy

2003

We study the size of the range of the derivatives of a smooth function between Banach spaces. We establish conditions on a pair of Banach spaces X and Y to ensure the existence of a C p smooth (Fréchet smooth or a continuous Gâteaux smooth) function f from X onto Y such that f vanishes outside a bounded set and all the derivatives of f are surjections. In particular we deduce the following results. For the Gâteaux case, when X and Y are separable and X is infinite-dimensional, there exists a continuous Gâteaux smooth function f from X to Y , with bounded support, so that f (X) = L(X, Y). In the Fréchet case, we get that if a Banach space X has a Fréchet smooth bump and dens X = dens L(X, Y), then there is a Fréchet smooth function f : X −→ Y with bounded support so that f (X) = L(X, Y). Moreover, we see that if X has a C p smooth bump with bounded derivatives and dens X = dens L m s (X; Y) then there exists another C p smooth function f : X −→ Y so that f (k) (X) = L k s (X; Y) for all k = 0, 1, ..., m. As an application, we show that every bounded starlike body on a separable Banach space X with a (Fréchet or Gâteaux) smooth bump can be uniformly approximated by smooth bounded starlike bodies whose cones of tangent hyperplanes fill the dual space X *. In the non-separable case, we prove that X has such property if X has smooth partitions of unity.