**Contemporary Mathematics**

Volume: 704;
2018;
290 pp;
Softcover

MSC: Primary 06; 11; 12; 30; 37; 40; 46; 47;

**Print ISBN: 978-1-4704-3491-5
Product Code: CONM/704**

List Price: $117.00

AMS Member Price: $93.60

MAA Member Price: $105.30

**Electronic ISBN: 978-1-4704-4676-5
Product Code: CONM/704.E**

List Price: $117.00

AMS Member Price: $93.60

MAA Member Price: $105.30

# Advances in Ultrametric Analysis

Share this page *Edited by *
*Alain Escassut; Cristina Perez-Garcia; Khodr Shamseddine*

This book contains the proceedings of the 14th International
Conference on \(p\)-adic Functional Analysis, held from June
30–July 4, 2016, at the Université d'Auvergne, Aurillac,
France.

Articles included in this book feature recent developments in
various areas of non-Archimedean analysis: summation of
\(p\)-adic series, rational maps on the projective line over
\(\mathbb{Q}p\), non-Archimedean Hahn-Banach theorems,
ultrametric Calkin algebras, \(G\)-modules with a convex base,
non-compact Trace class operators and Schatten-class operators in
\(p\)-adic Hilbert spaces, algebras of strictly differentiable
functions, inverse function theorem and mean value theorem in
Levi-Civita fields, ultrametric spectra of commutative non-unital
Banach rings, classes of non-Archimedean Köthe spaces,
\(p\)-adic Nevanlinna theory and applications, and sub-coordinate
representation of \(p\)-adic functions. Moreover, a paper on
the history of \(p\)-adic analysis with a comparative summary
of non-Archimedean fields is presented.

Through a combination of new research articles and a survey paper,
this book provides the reader with an overview of current developments
and techniques in non-Archimedean analysis as well as a broad
knowledge of some of the sub-areas of this exciting and
fast-developing research area.

#### Readership

Graduate students and research mathematicians interested in functional analysis, \(p\)-adic analysis, and \(p\)-adic dynamical systems.

# Table of Contents

## Advances in Ultrametric Analysis

- Cover Cover11
- Title page i2
- Contents iii4
- Preface v6
- Summary on non-Archimedean valued fields 18
- Introduction. 18
- 1. Ultrametric Spaces. 29
- - 1.1 The Strong Triangle Inequality and its Consequences. 29
- - 1.2 Compactness and Separability. 411
- - 1.3 Ultrametrizability. 411
- - 1.4 Spherical Completeness. 512
- 2. Valued fields. 714
- - 2.1 Archimedean and non-Archimedean valuations. 714
- - 2.2 Completion Theorem. 815
- - 2.3 Completion of Archimedean valued fields. 916
- - 2.4 Completion of non-Archimedean valued fields. 916
- - 2.5 Incompleteness of \QQ and πΎ(π₯). 1219
- - 2.6 Compact and Locally compact valued fields. 1421
- 3. Ordered fields. 1421
- - 3.1 Formally real fields. 1421
- - 3.2 General Hahn fields and the Embedding theorem. 1522
- - 3.3 Hahn Fields and Levi-Civita fields. 1724
- - 3.4 Real-closed field extensions of \RR. 1825
- 4. Algebraic closure of valued fields and their completions. 2027
- - 4.1 General Results. 2027
- - 4.2 Examples. 2128
- 5. General valuations. 2330
- 6. Maximal, Pseudo and Spherical completeness. 2633
- - 6.1 Pseudo-completeness. 2633
- - 6.2 Spherical completeness. 2633
- - 6.3 Maximal completeness. 2734
- 7. Structures of Maximally Complete fields. 2936
- - 7.1 Equal characteristic case. 2936
- - 7.2 Mixed characteristic case. 3037
- 8. Catalog of fields. 3138
- 9. Classification of fields. 3340
- References 3542

- Non-compact Trace class operators and Schatten-class operators in π-adic Hilbert spaces 3744
- 1. Introduction 3744
- 2. Background 3845
- 3. Schatten-class Operators in Classical Hilbert spaces 4047
- 4. Schatten Class operators in π-adic Hilbert spaces 4047
- 5. Completely Continuous operators and Schatten-class operators 4350
- 6. Trace and Schatten-class operators 4451
- Acknowledgments 4754
- References 4754

- Calculus on a non-Archimedean field extension of the real numbers: inverse function theorem, intermediate value theorem and mean value theorem 4956
- π-adic meromorphic functions πβπβ(π),πβπβ(π) sharing a small function, ignoring multiplicity 6976
- Spectra of commutative non-unital Banach rings 9198
- Ultrametric Calkin algebras 111118
- 1. The completely continuous linear operators 111118
- 2. Ultrametric Calkin algebras 114121
- 3. Ultrametric ultraproducts of Banach spaces 117124
- 4. A linear representation of the Calkin algebra when the Banach space πΈ is infinite dimensional of countable type 119126
- 5. Some additional remarks 122129
- References 124131

- On summation of π-adic series 127134
- Spectrum of ultrametric Banach algebras of strictly differentiable functions 139146
- Classical π-adic Nevanlinna theory and Nevalinna Theory out of a hole 161168
- The distance preserving mappings and isometrics defined on non-Archimedean Banach spaces 205212
- Rational map ππ₯+1/π₯ on the projective line over β_{π‘} 217224
- Non-Archimedean Hahn-Banach theorems and injective Banach spaces 231238
- πΊ-modules with a convex base 251258
- A journey throughout the history of π-adic numbers 261268
- On some classes of non-Archimedean KΓΆthe spaces 273280
- On the sub-coordinate representation of π-adic functions 285292
- Back Cover Back Cover1298