Introduction to Operator Theory in Riesz Spaces

✍ Scribed by Adriaan C. Zaanen

• Publisher: Springer
• Year: 1997
• Tongue: English
• Leaves: 324
• Edition: 1
• Category: Library

✦ Synopsis

The book deals with the structure of vector lattices, i.e. Riesz spaces, and Banach lattices, as well as with operators in these spaces. The methods used are kept as simple as possible. Almost no prior knowledge of functional analysis is required. For most applications some familiarity with the ordinary Lebesgue integral is already sufficient. In this respect the book differs from other books on the subject. In most books on functional analysis (even excellent ones) Riesz spaces, Banach lattices and positive operators are mentioned only briefly, or even not at all. The present book shows how these subjects can be treated without undue extra effort. Many of the results in the book were not yet known thirty years ago; some even were even not known then years ago.

✦ Table of Contents

Title Page......Page 3
Copyright Information......Page 4
Preface......Page 5
Contents......Page 9
1 Partially Ordered Sets......Page 13
2 Lattices......Page 16
3 Boolean Algebras......Page 19
4 Riesz Spaces......Page 25
5 Equalities and Inequalities......Page 29
6 Distributive Laws, the Birkhoff Inequalities and the Riesz Decomposition Property......Page 33
7 Ideals and Bands......Page 39
8 Disjointness......Page 46
9 Archimedean Riesz spaces......Page 51
10 Order Convergence and Uniform Convergence......Page 58
11 Projection Bands......Page 67
12 Dedekind Completeness......Page 73
13 Complex Riesz spaces......Page 83
14 Normed Spaces and Banach Spaces......Page 95
15 Normed Riesz Spaces and Banach Lattices......Page 97
16 The Riesz-Fischer Property......Page 111
17 Order Continuous Norms......Page 115
18 Linear Operators in Normed Spaces and in Riesz Spaces......Page 131
19 Riesz Homomorphisms and Quotient Spaces......Page 136
20 Order Bounded Operators......Page 145
21 Order Continuous Operators......Page 157
22 The Band of Order Continuous Operators......Page 162
23 Order Denseness......Page 173
24 The Carrier of an Operator......Page 176
25 The Order Dual of a Riesz Space......Page 181
26 Adjoint Operators......Page 189
27 The Space of Signed Measures......Page 195
28 The Radon-Nikodym Theorem......Page 200
29 Linear Functionals on Spaces of Measurable Functions......Page 205
30 Annihilators and Inverse Annihilators......Page 213
31 Embedding into the Order Bidual......Page 216
32 Projection Bands and Components......Page 221
33 Freudenthal's Spectral Theorem......Page 227
34 Functional Calculus......Page 233
35 Multiplication......Page 240
36 Complex Operators......Page 245
37 Synnatschke's Theorem......Page 252
38 The Hahn-Banach Theorem in Normed Vector Spaces......Page 255
39 The Hahn-Banach Theorem in Normed Riesz Spaces......Page 260
40 Spectrum and Resolvent Set......Page 265
41 The Krein-Rutman Theorem......Page 281
42 Irreducible Operators......Page 289
43 The Spectrum of a Compact Irreducible Operator......Page 298
44 The Peripheral Spectrum of a Positive Operator......Page 307
Index......Page 321