Non-Archimedean Valued Fields

Abstract

In the classical settings of the field of complex numbers \( \mathbb{C} \) and the field of real numbers \( \mathbb{R} \), the absolute value plays an important role in the Topology and in the Analysis on objects over these fields. In this chapter, we generalize the absolute value by introducing the notion of valuation on a general field \( \mathbb{K} \). As we shall see, this notion of valuation allows one to have a natural topology on the field itself and also on objects that are defined over the field. Analysis on the field and on these objects follows naturally. As it turns out, there are two kinds of valuation, one is the archimedean valuation, as in the cases of \( \mathbb{C} \) and \( \mathbb{R} \), and the other is the non-archimedean valuation. In this book, our focus will be on the non-archimedean valuation. More specifically, we will work on free Banach spaces over non-archimedean valued fields and Operator theory on them. In this chapter we shall first develop the theory of valuation and then we shall give many examples to illustrate the theory. This chapter will mostly serve as background for the theory of operators that will be developed later in the book. Most of the results are well-known and we gather only those that will serve our purposes.