𝔖 Bobbio Scriptorium
✦   LIBER   ✦

Hurewicz's Theorems and Renormings of Banach Spaces

✍ Scribed by Benoı̂t Bossard; Gilles Godefroy; Robert Kaufman

Publisher
Elsevier Science
Year
1996
Tongue
English
Weight
416 KB
Volume
140
Category
Article
ISSN
0022-1236

No coin nor oath required. For personal study only.

✦ Synopsis

Let N(X) be the set of all equivalent norms on a separable Banach space X, equipped with the topology of uniform convergence on bounded subsets of X. We show that if X is infinite dimensional, the set of all locally uniformly rotund norms on X reduces every coanalytic set and, thus, is in particular non-Borel. Dually, we show the same result for the set of all continuously differentiable norms on X, under the assumption X* is separable. This provides an analogue to a classical result of Mazurkiewicz within convex analysis.

📜 SIMILAR VOLUMES

Rolle's Theorem and Negligibility of Poi
Rolle's Theorem and Negligibility of Points in Infinite Dimensional Banach Spaces
✍ D Azagra; J Gómez; J.A Jaramillo 📂 Article 📅 1997 🏛 Elsevier Science 🌐 English ⚖ 198 KB

In this note we prove that if a differentiable function oscillates between y and on the boundary of the unit ball then there exists a point in the interior of the ball in which the differential of the function has norm equal or less than . This kind of approximate Rolle's theorem is interesting beca

The Failure of Rolle's Theorem in Infini
The Failure of Rolle's Theorem in Infinite-Dimensional Banach Spaces
✍ Daniel Azagra; Mar Jiménez-Sevilla 📂 Article 📅 2001 🏛 Elsevier Science 🌐 English ⚖ 171 KB

We prove the following new characterization of C p (Lipschitz) smoothness in Banach spaces. An infinite-dimensional Banach space X has a C p smooth (Lipschitz) bump function if and only if it has another C p smooth (Lipschitz) bump function f such that its derivative does not vanish at any point in

Inverse and Implicit Function Theorems f
Inverse and Implicit Function Theorems for Nonsmooth Maps in Banach Spaces
✍ Zsolt Páles 📂 Article 📅 1997 🏛 Elsevier Science 🌐 English ⚖ 273 KB

We extend the classical inverse and implicit function theorems, the implicit function theorems of Lyusternik and Graves, and the results of Clarke and Pourciau to the situation when the given function is not smooth, but it has a convex strict prederivative whose measure of noncompactness is smaller