Schwartz's theorem on mean periodic vector-valued functions
The structure of the section space of a real analytic vector bundle on a real analytic manifold X is studied. This is used to improve a result of Grothendieck and Poly on the zero spaces of elliptic operators and to extend a result of Domański and the author on the non-existence of bases to the present case.
Soit un espace riemannien symétrique et l’espace des fonctions continues sur tendant vers 0 à l’infini. On démontre qu’un opérateur , invariant par les isométries de , engendre un semi-groupe fortement continu de contractions sur s’il est dissipatif et si son domaine contient les fonctions de classe à support compact.
Several equivalent conditions are given for the existence of real-valued Baire functions of all classes on a type of -analytic spaces, called disjoint analytic spaces, and on all pseudocompact spaces. The sequential stability index for the Banach space of bounded continuous real-valued functions on these spaces is shown to be either , or (the first uncountable ordinal). In contrast, the space of bounded real-valued Baire functions of class 1 is shown to contain closed linear subspaces with index...
It is shown that complemented subspaces of s, that is, nuclear Fréchet spaces with properties (DN) and (Ω), which are 'almost normwise isomorphic' to a multiple direct sum of copies of themselves are isomorphic to s. This is applied, for instance, to spaces of Whitney jets on the Cantor set or the Sierpiński triangle and gives new results and also sheds new light on known results.