Transcription of Functional Analysis - Lancaster
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Functional Analysis - Lancaster
Functional Analysis Alexander C. R. Belton c Alexander C. R. Belton 2004, 2006. Copyright Hyperlinked and revised edition All rights reserved The right of Alexander Belton to be identified as the author of this work has been asserted by him in accordance with the Copyright, Designs and Patents Act 1988. Contents Contents i Introduction iii 1 Normed Spaces 3. Basic Definitions .. 3. Subspaces and Quotient Spaces .. 4. Completions .. 6. Direct Sums .. 7. Initial Topologies .. 9. Nets .. 10. Exercises 1 .. 14. 2 Linear Operators 17. Preliminaries .. 17. Completeness of B(X, Y ) .. 18. Extension of Linear Operators .. 18. The Baire Category Theorem .. 19. The Open-Mapping Theorem .. 21. Exercises 2 .. 22. The Closed-Graph Theorem .. 26. The Principle of Uniform Boundedness .. 27. The Strong Operator Topology .. 27. Exercises 3 .. 28. 3 Dual Spaces 31. Initial Definitions .. 31. The Weak Topology.
Introduction These notes are an expanded version of a set written for a course given to final-year undergraduates at the University of Oxford. A thorough understanding of the Oxford third-year b4 analysis course (an introduction
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