Banach Gelfand Triples for Applications in Physics and Engineering

The principle of extension is widespread within mathematics. Starting from simple objects one constructs more sophisticated ones, with a kind of natural embedding from the set of old objects to the new, enlarged set. Usually a set of operations on the old set can still be carried out, but maybe also some new ones. Done properly one obtains more completed objects of a similar kind, with additional useful properties. Let us give a simple example: While multiplication and addition can be done exactly and perfectly in the setting of Q, the rational numbers, the field R of real numbers has the advantage of being complete (Cauchy sequences have a limit...) and hence allowing for numbers like pi or 2. Finally the even “more complicated ” field C of complex numbers allows to find solutions to equations like z2 = −1. The chain of inclusions of fields, Q ⊂ R ⊂ C is a good motivating example in the domain of “numbers”. The main subject of the present survey-type article is a new theory of Banach Gelfand triples (BGTs), providing a similar setting in the context of (generalized) functions. Test functions are the simple objects, elements of the Hilbert space L2(Rd) are well suited in order to describe concepts of orthogonality, and they can be approximated to any given precision (in the ‖·‖2-norm