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Format: 1 online resource (527 pages)

ISBN: 0-08-095183-X

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Note: Bibliographic Level Mode of Issuance: Monograph , Front Cover -- Riesz Spaces -- Copyright Page -- Table of Contents -- Preface -- CHAPTER 1. DISTRIBUTIVE LATTICES AND NORMED FUNCTION SPACES -- 1. Partial ordering -- 2. Lattices -- 3. Boolean algebras -- 4. Boolean rings -- 5. Prime ideals -- 6. Hulls and kernels -- Stone's representation theorem -- 7. The hull-kernel topology -- 8. Compactness and separation properties of the hull-kernel topology -- 9. Nörmed Kothe spaces -- 10. Orlicz spaces -- CHAPTER 2. ELEMENTARY PROPERTIES OF ORDERED VECTOR SPACES AND RIESZ SPACES -- 11. Ordered vector spaces and Riesz spaces -- 12. Inequalities and distributive laws in a Riesz space -- 13. Suprema and infima of subsets of an ordered vector space -- 14. Disjointness -- 15. Monotone sequences and directed sets -- 16. Order convergence and relatively uniform convergence -- CHAPTER 3. IDEALS AND BANDS -- ARCHIMEDEAN RIESZ SPACES -- 17. Riesz subspaces, ideals and bands -- 18. Riesz homomorphisms and quotient spaces -- 19. Disjoint complements -- 20. The band generated by an ideal -- 21. Order dense ideals, quasi order dense ideals and super order dense ideals -- 22. Archimedean Riesz spaces -- CHAPTER 4. DEDEKIND COMPLETENESS AND PROJECTIONP ROPERTIES -- 23. Dedekind completeness -- 24. Projection properties -- 25. The main inclusion theorem -- 26. Atoms and finitedimensional Riesz spaces -- 27. Maximal ideals -- 28. Order bases and disjoint systems -- 29. Bands or ideals with a countable order basis and the relation with order separability -- 30. The Boolean algebra of all projection bands -- 31. Principal projection bands -- 32. Dedekind completion of an Archimedean Riesz space -- CHAPTER 5. PRIME IDEALS -- 33. Prime ideals -- 34. Prime ideals in C(X) for X compact and Hausdorff -- 35. Hulls and kernels -- 36. The hull-kernel topology -- 37. Compactness and separation properties of the hull-kernel topology. , CHAPTER 6. FREUDENTHAL'S SPECTRAL THEOREM -- 38. Spectral systems -- 39. Uniform Cauchy sequences -- 40. Freudenthal's spectral theorem -- 41. The Radon-Nikodym theorem and the Poisson formula -- 42. Uniform completeness, Dedekind completeness and the spectral heorem -- 43. The spaces C(X) and Cb(X) -- CHAPTER 7. SPECTRAL REPRESENTATION THEORY -- 44. The Johnson-Kist representation theorem -- 45. The Yosida spectral representation theorem in an Archimedean Riesz space with strong unit -- 46. The Yosida spectral representation theorem in an Archimedean Riesz space -- 47. The space C∞(X) for X extremally disconnected -- 48. Subsets of the set of all bands -- 49. The Ogasawara-Maeda representation theorem -- 50. The Ogasawara-Maeda representation theorem for an Archimedean space with a weak unit or with a disjoint order basis -- 51. The Nakano representation theorem -- 52. Prime ideal extension in a Riesz space -- CHAPTER 8. HERMITIAN OPERATORS IN HILBERT SPACE -- 53. The ordered vector space of Hermitian operators in a Hilbert space -- 54. The square root of a positive Hermitian operator -- 55. The second commutant of a commuting set of Hermitian operators -- 56. The spectral theorem for Hermitian operators and normal operators -- 57. The structure of the second commutant -- 58. Kadison's anti-lattice theorem -- CHAPTER 9. RIESZ HOMOMORPHISMS AND QUOTIENT SPACES -- 59. Additional results on Riesz homomorphisms -- 60. Archimedean quotient spaces -- 61. Riesz spaces with the property that every quotient space is Archimedeandean -- 62. The ideal of the infinitely small elements and Riesz seminorms with their quotient seminorms, in particular in spaces with a strong unit -- 63. Riesz homomorphisms and the relatively uniform topology -- 64. Images of principal bands under a Riesz homomorphism -- 65. Riesz σ-homomorphisms and the order topology. , 66. Normal Riesz homomorphisms -- CHAPTER 10. THE EGOROFF PROPERTY AND THE DIAGONAL PROPERTY -- 67. The Egoroff property -- 68. The diagonal property -- 69. The Egoroff property and the diagonal property in order bounded sets -- 70. Stability of order convergence and regular Riesz spaces -- 71. Function spaces possessing the diagonal property -- 72. The diagonal property for relatively uniform convergence -- 73. The Egoroff property in Boolean rings -- 74. Egoroff's theorem -- 75. The Egoroff property and Dedekind completeness -- BIBLIOGRAPHY -- SUBJECT INDEX. , English

Additional Edition: ISBN 0-444-10129-2

Language: English