Analysis on a Nonarchimedean Extension of the Real Numbers Lecture Notes

A field extension R of the real numbers is presented. It has similar algebraic properties as ; for example, all roots of positive numbers exist, and the structure C obtained by adjoining the imaginary unit is algebraically complete. The set can be totally ordered and contains infinitely small and infinitely large quantities. Under the topology induced by the ordering, the set becomes Cauchy complete; but different from , there is a second natural way of introducing a topology. It is shown that R is the smallest totally ordered algebraically complete extension of Power series have identical convergence properties as in , and thus important transcendental functions exist and behave as in . Furthermore, there is a natural way to extend any other real function under preservation of its smoothness prop-erties. In addition to these common functions, delta functions can be introduced directly. A calculus involving continuity, differentiability and integrability is developed. Central concepts like the intermediate value theorem, mean value theorem, an