Intersections of Fréchet Spaces and (LB)-Spaces

Abstract

This article presents results about the class of locally convex spaces which are defined as the intersection E∩F of a Frechet space F and a countable inductive limit of Banach spaces E. This class appears naturally in analytic applications to linear partial differential operators. The intersection has two natural topologies, the intersection topology and an inductive limit topology. The first one is easier to describe and the second one has good locally convex properties. The coincidence of these topologies and its consequences for the spaces E ∩ F and E + F are investigated. 2000 Mathematics Subject Classification. Primary: 46A04, secondary: 46A08, 46A13, 46A45. The aim of this paper is to investigate spaces E ∩ F which are the intersection of a Frechet space F and an (LB)-space E. They appear in several parts of analysis whenever the space F is determined by countably many necessary (e.g. differentiability of integrability) conditions and E is the dual of such a space, in particular E is defined by a countable sequence of bounded sets which may also be determined by concrete estimates. Two natural topologies can be defined on E ∩ F : the intersection topology, which has seminorms easy to describe and which permits direct estimates, and a finer inductive limit topology which is defined in a natural way and which has good locally convex properties, e.g. E ∩F with this topology is a barrelled space. It is important to know when these two topologies coincide. It turns out that the locally convex properties of E ∩ F with the intersection topology are related to the completeness of the (LF)-space E + F . The present setting provides us with new interesting examples of (LF)-spaces. Recent progress on the study of (LF)-spaces, see [4, 14, 25, 27], is very important in our work. Our main results are Theorems 4 and 7. The Examples 5 and 10 show the main difficulties. Our article continues the research in [5], where the case when F is a Frechet Schwartz space (an (FS)-space) and E is the dual of an (FS)-space (a (DFS)-space) was analyzed. One of the original motivations for the research in [5] came up in the investigations of Langenbruch [18, 19, 20] about the surjectivity of linear partial differential operators P (D) with constant coefficients on spaces of ultradifferentiable functions and ultradistributions. The extension of {ω}-ultradifferentiable regularity for {ω}ultradistributions u such that P (D)u = 0 across a hypersurface has consequences for the surjectivity of P (D), in the same way that Holmgren’s uniqueness theorem is relevant in the discussions of surjectivity problems for P (D) on C∞(Ω), see [15, Chapters 8 and 10]. The extension of regularity for ultradistributions was studied by Langenbruch in [18]. In the classical case, treated by Hormander in [15, Section 11.3], one seeks conditions on a linear subspace V of RN and on The research of Jose Bonet was partially supported by FEDER and MCYT Proyecto no. BFM20012670.