Robert Deville

Université Bordeaux-I · UMR IMB - Institut de Mathématiques de Bordeaux

Let X be a Banach space, K be a scattered compact and T: B C(K) → X be a Fréchet smooth operator whose derivative is uniformly continuous. We introduce the smooth biconjugate T**: B C(K)** → X** and prove that if T is noncompact, then the derivative of T** at some point is a noncompact linear operator. Using this we conclude, among other things, th...

We prove that there exists a Lipschitz function froml 1 into ℝ2 which is Gâteaux-differentiable at every point and such that for everyx, y εl 1, the norm off′(x) −f′(y) is bigger than 1. On the other hand, for every Lipschitz and Gâteaux-differentiable function from an arbitrary Banach spaceX into ℝ and for everyε > 0, there always exist two points...

We investigate the best order of smoothness of Lp(Lq). We prove in particular that there exists a [script C][infty infinity]-smooth bump function on Lp(Lq) if and only if p and q are even integers and p is a multiple of q.

A closed, convex and bounded setP in a Banach spaceE is called a polytope if every finite-dimensional section ofP is a polytope. A Banach spaceE is called polyhedral ifE has an equivalent norm such that its unit ball is a polytope. We prove here: (1) LetW be an arbitrary closed, convex and bounded body in a separable polyhedral Banach spaceE and...

We present a formula for the viscosity subdifferential of the sum of two uniformly continuous functions on smooth Banach spaces. This formula is deduced from a new variational principle with constraints. We obtain as a consequence a weak form of Preiss' theorem for uniformly continuous functions. We use these results to give simple proofs of some u...

We investigate the best order of smoothness of $L^p(L^q)$. We prove in particular that there exists a $C^\infty$-smooth bump function on $L^p(L^q)$ if and only if $p$ and $q$ are both even integers and $p$ is a multiple of $q$. Comment: 18 pages; AMS-Tex

It is well known that in separable Banach spaces, or more generally in WCD Banach spaces, the existence of a C k-Fréchet differentiable bump function implies the possibility of uniform approximation of continuous functions by C k-smooth functions. However, the more subtle question of the uniform approximation on bounded sets