From Banach to distributions and currents

Pierre Dolbeault
2009 Lvov Mathematical School in the Period 1915-45   unpublished
1. Introduction. The Banach spaces as described in the seminal book [B] are still actively studied in many domains, in particular the geometry of Banach spaces (see [JL]). Moreover, from the beginning of the '30's, Banach spaces with an order relation were introduced and their theory particularly developed in relation with the topology, around 1980 and until now (see [AA]). They were also used during the '30's where particular distributions were introduced by Sobolev (see Section 3). In the
more » ... tion 3). In the meantime, general distributions were defined and developed in the late '40's (Section 4), then it was necessary to introduce more general function and distribution spaces (Section 5). We propose to describe the structures of the function and distribution spaces with their relations with Banach spaces and their generalizations to differential forms and currents in global analysis (Sections 5-6). We shall see that the (norm) topology, the completeness, the operators, the dual, the adjoint of an operator will still be used in the generalized context. No attempt will be done to anymore enter in the intricate history of distributions.
doi:10.4064/bc87-0-9 fatcat:rly2pjuzdbezpdx54hhtzro4je