Tauberian operators were introduced to investigate a problem in summability theory from an abstract point of view. Since that introduction, they have made a deep impact on the isomorphic theory of Banach spaces. In fact, these operators have been useful in several contexts of Banach space theory that have no apparent or obvious connections. For instance, they appear in the famous factorization of Davis, Figiel, Johnson and Pelczynski [49] (henceforth the DFJP factorization), in the study of exact sequences of Banach spaces [174], in the solution of certain summability problems of tauberian type [63, 115], in the problem of the equivalence between the Krein-Milman property and the Radon-Nikodym property [151], in certain sequels of James' characterization of reflexive Banach spaces [135], in the construction of hereditarily indecomposable Banach spaces [13], in the extension of the principle of local reflexivity to operators [27], in the study of certain Calkin algebras associated with the weakly compact operators [16], etc. Since the results concerning tauberian operators appear scattered throughout the literature, in this book we give a unified presentation of their properties and their main applications in functional analysis. We also describe some questions about tauberian operators that remain open. This book has six chapters and an appendix. In Chapter 1 we show how the concept of tauberian operator was introduced in the study of a classical problem in summability theory - the characterization of conservative matrices that sum no bounded divergent sequences - by means of functional analysis techniques. One of those solutions is due to Crawford [45], who considered the second conjugate of the operator associated with one of those matrices.

A typical source of mistakes that frequently lead to a wrong or incomplete solution for the antiderivative of a given real function of one real variable is a misuse of the technique of change of variable. The increasing implementation of software in apparently mechanic tasks such as the calculation of antiderivatives has not improved the situation, yet those software packages issue generic warnings such as "the answer's is not guaranteed to be continuous" or "the solution might be only valid for parts of the function". The practical meaning of those vague machine messages is clearly envisaged in this book, which shows how to handle the technique of change of variable in order to provide correct solutions. This book is monographically focused on elementary antidifferentiation and reasonably self-contained, yet it is written in a "hand-book" style: it has plenty of examples and graphics in an increasing level of difficulty; the most standard changes of variable are studied and the hardest theoretic parts are included in a final Appendix. Each practical chapter has a list of exercises and solutions. This book is intended for instructors and university students of Mathematics of first and second year.