Monographical Booklets in applied & computer Mathematics MB-16

Előszó

Preface to the volume MB-16
The ultrametric normed (or non-archimedean valued) fields arise in Mathematics in relationship with the congruences in the Number Theory. But there arc more reasons to connect them with Analysis because of the existence of strong triangle inequality for distances, which provides the possibility to make new analogous constructions as those in the classical analysis. The results so obtained confer a new point of view on the Analysis itself, but these are of interest for many other domains, such as: the Functional Analysis, the Quantum Mechanics, the Differential Geometry and, recently, the Numerical Analysis. Each of them seems to be interested of the non-archimedean completion of the rational number field Q, regarded as an alternative to the usual completion of this field, which leads to the real number field and to its extensions.
It is well known that the Analysis over the fields R, C, M is of archimedean type, but the Analysis, and therefore all other mathematical domains based on it, over any field different from the former is by all means non-archimedean.
The main subjects concerning this last variant of mathematics in appearance are similar with those of the archimedean case, but the results and also the mathematical and physical properties of many things in a non-archimedean approach are nevertheless essentially different in such a case. In the non-archimedean mathematics there are many specific notions. One of them, the spherical completeness, plays a crucial role. This notion does not appear in the real or complex analysis. The property of a space or of a field to be spherically complete is a powerful property frequently used in Topology or in Functional Analysis of non-archimedean type. Many mathematicians studied it in the last decades of this century, such as: J.P.Serre, A.F.Monna. J.Martinez-Maurica. T.Pellón. W.H.Schikhof. J.M.Bayod, C.Pérez-Garcia, N.De Grande-De Kimpe, J.Kakol, J.Van TieL, and many others. A characterization of the spherical completeness of a non-archimedean complete non-trivially valued field K and of a Hausdorff locally convex space over such a field in terms of classical theorems of Functional Analysis is given. The most important example of a field with the specified above properties is that of p-adic numbers, which also can be obtained by the completion of Q with
respect to an ultrametric norm.
The p-adic number field Rp plays a special role in the study of non-archimedean analysis and geometry . It is an analogous in these studies of the real field IS used in archimedean analysis, functional analysis, geometry, physics, etc. Vissza