Some Useful Theorems

Abstract

In this chapter we collect a number of results from real analysis, which are useful to solve the exercises. The results presented are along one main theme: How to interchange two operations in analysis (for instance order of integration in a double integral, integration of a function depending on a parameter and derivation with respect to this parameter,...). Most, if not all, of the results, can be proved by elementary methods, but are also special cases of general theorems from the theory of integration (such as the dominated convergence theorem, Fubini’s theorem,...). Some aspects of this theory are reviewed in Chapter 15. Finally, note that we consider complex-valued functions. The results are easily derived in the complex case from their real counterparts. In fact, they are sometimes still valid for functions and sequences with values in a Banach space or a Banach algebra, but a discussion of this latter point is far outside the framework of this book.