Introduction to Operator Theory in Riesz Spaces

1. Front Matter

   Pages I-XI

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2. Lattices and Boolean Algebras

   • Adriaan C. Zaanen

   Pages 1-11

3. Riesz Spaces

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   Pages 13-26

4. Ideals, Bands and Disjointness

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   Pages 27-38

5. Archimedean Spaces and Convergence

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   Pages 39-53

6. Projections and Dedekind Completeness

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   Pages 55-70

7. Complex Riesz Spaces

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   Pages 71-81

8. Normed Riesz Spaces and Banach Lattices

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   Pages 83-97

9. The Riesz-Fischer Property and Order Continuous Norms

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   Pages 99-117

10. Linear Operators

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    Pages 119-131

11. Order Bounded Operators

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    Pages 133-143

12. Order Continuous Operators

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    Pages 145-160

13. Carriers of Operators

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    Pages 161-168

14. Order Duals and Adjoint Operators

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    Pages 169-181

15. Signed Measures and the Radon-Nikodym Theorem

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    Pages 183-192

16. Linear Functionals on Spaces of Measurable Functions

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    Pages 193-200

17. Embedding into the Bidual

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    Pages 201-208

18. Freudenthal’s Spectral Theorem

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    Pages 209-219

19. Functional Calculas and Multiplication

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    Pages 221-232

20. Complex Operators

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    Pages 233-241

Since the beginning of the thirties a considerable number of books on func­ tional analysis has been published. Among the first ones were those by M. H. Stone on Hilbert spaces and by S. Banach on linear operators, both from 1932. The amount of material in the field of functional analysis (in­ cluding operator theory) has grown to such an extent that it has become impossible now to include all of it in one book. This holds even more for text­ books. Therefore, authors of textbooks usually restrict themselves to normed spaces (or even to Hilbert space exclusively) and linear operators in these spaces. In more advanced texts Banach algebras and (or) topological vector spaces are sometimes included. It is only rarely, however, that the notion of order (partial order) is explicitly mentioned (even in more advanced exposi­ tions), although order structures occur in a natural manner in many examples (spaces of real continuous functions or spaces of measurable function~). This situation is somewhat surprising since there exist important and illuminating results for partially ordered vector spaces, in . particular for the case that the space is lattice ordered. Lattice ordered vector spaces are called vector lattices or Riesz spaces. The first results go back to F. Riesz (1929 and 1936), L. Kan­ torovitch (1935) and H. Freudenthal (1936).

• Boolean algebra
• Calc
• Excel
• Morphism
• Multiplication
• calculus
• convergence
• functional analysis
• functions
• homomorphism
• knowledge
• peripheral
• spectral theorem
• spectral theory
• vector lattice

• Delft, The Netherlands

  Adriaan C. Zaanen

• Department of Mathematics and Computer Sciences, University of Leiden, Leiden, The Netherlands

  Adriaan C. Zaanen

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