The purpose of this book is to present the main structure theorems in the isometric theory of classical Banach spaces. Elements of general topology, measure theory, and Banach spaces are assumed to be familiar to the reader. A classical Banach space is a Banach space X whose dual space is linearly isometric to Lp(j1, IR) (or Lp(j1, CC) in the complex case) for some measure j1 and some 1 ~ p ~ 00. If 1 < p < 00, then it is well known that X=L (j1,IR) where 1/p+1/q=1 and if p=oo, then X=L (v,lR) for q j some measure v. Thus, the only case where a space is obtained which is not truly classical is when p = 1. This class of spaces is known as L - 1 predual spaces since their duals are L type. It includes some well known j subclasses such as spaces of the type C(T, IR) for T a compact Hausdorff space and abstract M spaces. The structure theorems concern necessary and sufficient conditions that a general Banach space is linearly isometric to a classical Banach space. They are framed in terms of conditions on the norm of the space X, conditions on the dual space X*, and on (finite dimensional) subspaces of X. Since most of these spaces are Banach lattices and Banach algebras, characterizations among theses classes are also given.

Inhalt

1. Partially Ordered Banach Spaces.- § 1. Vector Lattices.- § 2. Partially Ordered Normed Linear Spaces.- § 3. Normed Linear Lattices.- 2. Some Aspects of Topology and Regular Borel Measures.- § 4. Existence Theorems for Continuous Functions.- § 5. Dispersed Compact Hausdorff Spaces.- § 6. The Cantor Set.- § 7. Extremally Disconnected Compact Hausdorff Spaces.- § 8. Regular Borel Measures.- 3. Characterizations of Banach Spaces of Continuous Functions.- § 9. Lattice and Algebraic Characterizations of Banach Spaces of Continuous Functions.- § 10. Banach Space Characterizations of Spaces of Continuous Functions.- § 11. Banach Spaces with the Hahn-Banach Extension Property.- 4. Classical Sequence Spaces.- § 12. Schauder Bases in Classical Sequence Spaces.- § 13. Embedding of Classical Sequence Spaces into Continuous Function Spaces.- 5. Representation Theorems for Spaces of the Type Lp(T, ?, µ, ?).- § 14. Measure Algebras and the Representation of Lp(T, ?, µ, ?) when µ a Finite Measure.- § 15. Abstract Lp Spaces.- 6. Characterizations of Abstract M and Lp Spaces.- § 16. Positive Contractive Projections in Abstract M and Lp Spaces.- § 17. Contractive Projections in Abstract Lp Spaces.- § 18. Geometric Properties of Abstract L1 Spaces and Some Dual Abstract L1 Spaces.- 7. L1-Predual Spaces.- § 19. Partially Ordered L1-Predual Spaces.- § 20. Compact Choquet Simplexes.- § 21. Characterizations of Real L1-Predual Spaces.- § 22. Some Selection and Embedding Theorems for Real L1-Predual Spaces.- § 23. Characterizations of Complex L1-Predual Spaces.