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Short description/annotation
This is a short course on Banach space theory applicable to many contemporary research arenas.
Main description
This is a short course on Banach space theory with special emphasis on certain aspects of the classical theory. In particular, the course focuses on three major topics: The elementary theory of Schauder bases, an introduction to Lp spaces, and an introduction to C(K) spaces. While these topics can be traced back to Banach himself, our primary interest is in the postwar renaissance of Banach space theory brought about by James, Lindenstrauss, Mazur, Namioka, Pelczynski, and others. Their elegant and insightful results are useful in many contemporary research endeavors and deserve greater publicity. By way of prerequisites, the reader will need an elementary understanding of functional analysis and at least a passing familiarity with abstract measure theory. An introductory course in topology would also be helpful; however, the text includes a brief appendix on the topology needed for the course.
Table of contents:
Preface; 1. Classical Banach spaces; 2. Preliminaries; 3. Bases in Banach spaces; 4. Bases in Banach spaces II; 5. Bases in Banach spaces III; 6. Special properties of C0, l1, and l; 7. Bases and duality; 8. Lp spaces; 9. Lp spaces II; 10. Lp spaces III; 11. Convexity; 12. C(K) Spaces; 13. Weak compactness in L1; 14. The Dunford-Pettis property; 15. C(K) Spaces II; 16. C(K) Spaces III; A. Topology review.
This is a short course on Banach space theory applicable to many contemporary research arenas.
Main description
This is a short course on Banach space theory with special emphasis on certain aspects of the classical theory. In particular, the course focuses on three major topics: The elementary theory of Schauder bases, an introduction to Lp spaces, and an introduction to C(K) spaces. While these topics can be traced back to Banach himself, our primary interest is in the postwar renaissance of Banach space theory brought about by James, Lindenstrauss, Mazur, Namioka, Pelczynski, and others. Their elegant and insightful results are useful in many contemporary research endeavors and deserve greater publicity. By way of prerequisites, the reader will need an elementary understanding of functional analysis and at least a passing familiarity with abstract measure theory. An introductory course in topology would also be helpful; however, the text includes a brief appendix on the topology needed for the course.
Table of contents:
Preface; 1. Classical Banach spaces; 2. Preliminaries; 3. Bases in Banach spaces; 4. Bases in Banach spaces II; 5. Bases in Banach spaces III; 6. Special properties of C0, l1, and l; 7. Bases and duality; 8. Lp spaces; 9. Lp spaces II; 10. Lp spaces III; 11. Convexity; 12. C(K) Spaces; 13. Weak compactness in L1; 14. The Dunford-Pettis property; 15. C(K) Spaces II; 16. C(K) Spaces III; A. Topology review.