10.4 Miguel Garcia Bravo (Universidad Complutense de Madrid): Approximate Morse-Sard type results in infinite dimensional Banach spaces

Abstract: The classical Morse theorem from 1939 establishes that if f: R^n--->R is of class C^n then we can ensure that its set of critical values has Lebesgue measure zero. We will raise the question of what happens in infinite dimensional Banach spaces. Unfortunately there are C^{infty} functions from the separable infinite dimensional Hilbert space into R whose set of critical values is the unit interval. However, one can prove approximate versions of the Morse-Sard theorem in the following way: any given continuous function can be uniformly approximated by a smooth one without any critical point. This type of result is counterintuitive, as it is false in finite dimensions. This is as well related to the failure of Rolle's theorem in infinite dimensions.
The talk is supposed to be an introduction to all this theory and, whenever possible, some of the keys behind the proofs will be given.