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An Introduction to Functional Analysis

  • Date Published: March 2020
  • availability: Available
  • format: Paperback
  • isbn: 9780521728393


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About the Authors
  • This accessible text covers key results in functional analysis that are essential for further study in the calculus of variations, analysis, dynamical systems, and the theory of partial differential equations. The treatment of Hilbert spaces covers the topics required to prove the Hilbert–Schmidt theorem, including orthonormal bases, the Riesz representation theorem, and the basics of spectral theory. The material on Banach spaces and their duals includes the Hahn–Banach theorem, the Krein–Milman theorem, and results based on the Baire category theorem, before culminating in a proof of sequential weak compactness in reflexive spaces. Arguments are presented in detail, and more than 200 fully-worked exercises are included to provide practice applying techniques and ideas beyond the major theorems. Familiarity with the basic theory of vector spaces and point-set topology is assumed, but knowledge of measure theory is not required, making this book ideal for upper undergraduate-level and beginning graduate-level courses.

    • Includes an extensive source of homework problems for instructors and independent study
    • Presents functional analytical methods without a reliance on measure-theoretic results, making the topics more widely accessible
    • Provides readers with a sense of accomplishment and closure by showing how both Hilbert space theory and Banach space theory aim towards major results with important applications
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    Reviews & endorsements

    'This excellent introduction to functional analysis brings the reader at a gentle pace from a rudimentary acquaintance with analysis to a command of the subject sufficient, for example, to start a rigorous study of partial differential equations. The choice and order of topics are very well thought-out, and there is a fine balance between general results and concrete examples and applications.' Charles Fefferman, Princeton University, New Jersey

    'An Introduction to Functional Analysis covers everything that one would expect to meet in an undergraduate course on this elegant area and more, including spectral theory, the category-based theorems and unbounded operators. With a well-written narrative and clear detailed proofs, together with plentiful examples and exercises, this is both an excellent course book and a valuable reference for those encountering functional analysis from across mathematics and science.' Kenneth Falconer, University of St Andrews, Scotland

    'This is a beautifully written book, containing a wealth of worked examples and exercises, covering the core of the theory of Banach and Hilbert spaces. The book will be of particular interest to those wishing to learn the basic functional analytic tools for the mathematical analysis of partial differential equations and the calculus of variations.' Endre Suli, University of Oxford

    '… this is a valuable book. It is an accessible yet serious look at the subject, and anybody who has worked through it will be rewarded with a good understanding of functional analysis, and should be in a position to read more advanced books with profit.' Mark Hunacek, The Mathematical Gazette

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    Product details

    • Date Published: March 2020
    • format: Paperback
    • isbn: 9780521728393
    • length: 416 pages
    • dimensions: 227 x 153 x 22 mm
    • weight: 0.6kg
    • contains: 17 b/w illus. 215 exercises
    • availability: Available
  • Table of Contents

    Part I. Preliminaries:
    1. Vector spaces and bases
    2. Metric spaces
    Part II. Normed Linear Spaces:
    3. Norms and normed spaces
    4. Complete normed spaces
    5. Finite-dimensional normed spaces
    6. Spaces of continuous functions
    7. Completions and the Lebesgue spaces Lp(Ω)
    Part III. Hilbert Spaces:
    8. Hilbert spaces
    9. Orthonormal sets and orthonormal bases for Hilbert spaces
    10. Closest points and approximation
    11. Linear maps between normed spaces
    12. Dual spaces and the Riesz representation theorem
    13. The Hilbert adjoint of a linear operator
    14. The spectrum of a bounded linear operator
    15. Compact linear operators
    16. The Hilbert–Schmidt theorem
    17. Application: Sturm–Liouville problems
    Part IV. Banach Spaces:
    18. Dual spaces of Banach spaces
    19. The Hahn–Banach theorem
    20. Some applications of the Hahn–Banach theorem
    21. Convex subsets of Banach spaces
    22. The principle of uniform boundedness
    23. The open mapping, inverse mapping, and closed graph theorems
    24. Spectral theory for compact operators
    25. Unbounded operators on Hilbert spaces
    26. Reflexive spaces
    27. Weak and weak-* convergence
    Appendix A. Zorn's lemma
    Appendix B. Lebesgue integration
    Appendix C. The Banach–Alaoglu theorem
    Solutions to exercises

  • Author

    James C. Robinson, University of Warwick
    James C. Robinson is a professor in the Mathematics Institute at the University of Warwick. He has been the recipient of a Royal Society University Research Fellowship and an Engineering and Physical Sciences Research Council (EPSRC) Leadership Fellowship. He has written six books in addition to his many publications in infinite-dimensional dynamical systems, dimension theory, and partial differential equations.

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