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Preface

Published online by Cambridge University Press:  12 January 2010

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Summary

This elementary book is intended for advanced undergraduates or anyone on a higher level who wants to learn the basic facts of p-adic analysis. We only assume the reader to have some standard knowledge of analysis and algebra.

In analysis (and outside it) the fields ℝ and ℂ play a central role. For several reasons people started to study the implications of replacing ℝ or ℂ by a more general object, viz. a field K with a complete valuation │ │ comparable to the absolute value function (see Definition 1.1). Many such fields other than ℝ or ℂ exist, their valuations are all ‘non-archimedean’, i.e. they satisfy the ‘strong triangle inequality’ │x + y│ ≤ max(│x│, │y│). The analysis in and over non-archimedean valued fields K is known as ultrametric (non-archimedean, p-adic) analysis.

In this book we shall treat the basic facts of ultrametric analysis together to form an alternative ‘one variable calculus course’. Thus, in K we shall consider familiar concepts such as continuity, differentiability, (power) series, integration, etc. However, the strong triangle inequality causes fascinating deviations from the ‘classical analysis’ (over ℝ or ℂ); let us mention a few of them.

  1. (i) A series Σan in K converges if limn → ∞an = 0. The power series Σxn/n! of exp (if it makes sense at all) converges only on a disc strictly contained in the closed unit disc {x : │x│ ≤ 1}. Hence Σ1/n! diverges (but Σn! converges in many K).

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Ultrametric Calculus
An Introduction to p-Adic Analysis
, pp. ix - xii
Publisher: Cambridge University Press
Print publication year: 1985

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  • Preface
  • W. H. Schikhof
  • Book: Ultrametric Calculus
  • Online publication: 12 January 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511623844.001
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  • Preface
  • W. H. Schikhof
  • Book: Ultrametric Calculus
  • Online publication: 12 January 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511623844.001
Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • Preface
  • W. H. Schikhof
  • Book: Ultrametric Calculus
  • Online publication: 12 January 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511623844.001
Available formats
×