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AuthorHolmes, Richard B. author
TitleGeometric Functional Analysis and its Applications [electronic resource] / by Richard B. Holmes
ImprintNew York, NY : Springer New York, 1975
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Descript X, 246 p. online resource


This book has evolved from my experience over the past decade in teaching and doing research in functional analysis and certain of its appliยญ cations. These applications are to optimization theory in general and to best approximation theory in particular. The geometric nature of the subjects has greatly influenced the approach to functional analysis presented herein, especially its basis on the unifying concept of convexity. Most of the major theorems either concern or depend on properties of convex sets; the others generally pertain to conjugate spaces or compactness properties, both of which topics are important for the proper setting and resolution of optimization problems. In consequence, and in contrast to most other treatments of functional analysis, there is no discussion of spectral theory, and only the most basic and general properties of linear operators are established. Some of the theoretical highlights of the book are the Banach space theorems associated with the names of Dixmier, Krein, James, Smulian, Bishop-Phelps, Brondsted-Rockafellar, and Bessaga-Pelczynski. Prior to these (and others) we establish to two most important principles of geometric functional analysis: the extended Krein-Milman theorem and the Hahnยญ Banach principle, the latter appearing in ten different but equivalent formulaยญ tions (some of which are optimality criteria for convex programs). In addition, a good deal of attention is paid to properties and characterizations of conjugate spaces, especially reflexive spaces


I Convexity in Linear Spaces -- ยง 1. Linear Spaces -- ยง 2. Convex Sets -- ยง 3. Convex Functions -- ยง 4. Basic Separation Theorems -- ยง 5. Cones and Orderings -- ยง 6. Alternate Formulations of the Separation Principle -- ยง 7. Some Applications -- ยง 8. Extremal Sets -- Exercises -- II Convexity in Linear Topological Spaces -- ยง 9. Linear Topological Spaces -- ยง10. Locally Convex Spaces -- ยง11. Convexity and Topology -- ยง12. Weak Topologies -- ยง13. Extreme Points -- ยง14. Convex Functions and Optimization -- ยง15. Some More Applications -- Exercises -- III Principles of Banach Spaces -- ยง16. Completion, Congruence, and Reflexivity -- ยง17. The Category Theorems -- ยง18. The Smulian Theorems -- ยง19. The Theorem of James -- ยง20. Support Points and Smooth Points -- ยง21. Some Further Applications -- Exercises -- IV Conjugate Spaces and Universal Spaces -- ยง22. The Conjugate of C(?, ?) -- ยง23. Properties and Characterizations of Conjugate Spaces -- ยง24. Isomorphism of Certain Conjugate Spaces -- ยง25. Universal Spaces -- Exercises -- References -- Symbol Index

Mathematics Mathematical analysis Analysis (Mathematics) Mathematics Analysis


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