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A Hilbert Space Problem Book (Graduate Texts in Mathematics) [Hardcover]

P.R. Halmos
4.5 out of 5 stars  See all reviews (2 customer reviews)

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Book Description

December 1, 1982 0387906851 978-0387906850 2nd rev. and enlarged ed. 1982

From the Preface: "This book was written for the active reader. The first part consists of problems, frequently preceded by definitions and motivation, and sometimes followed by corollaries and historical remarks... The second part, a very short one, consists of hints... The third part, the longest, consists of solutions: proofs, answers, or contructions, depending on the nature of the problem....

This is not an introduction to Hilbert space theory. Some knowledge of that subject is a prerequisite: at the very least, a study of the elements of Hilbert space theory should proceed concurrently with the reading of this book."


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A Hilbert Space Problem Book (Graduate Texts in Mathematics) + Introduction to Hilbert Space and the Theory of Spectral Multiplicity
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Product Details

  • Series: Graduate Texts in Mathematics (Book 19)
  • Hardcover: 396 pages
  • Publisher: Springer; 2nd rev. and enlarged ed. 1982 edition (December 1, 1982)
  • Language: English
  • ISBN-10: 0387906851
  • ISBN-13: 978-0387906850
  • Product Dimensions: 0.9 x 6.1 x 9.1 inches
  • Shipping Weight: 1.3 pounds (View shipping rates and policies)
  • Average Customer Review: 4.5 out of 5 stars  See all reviews (2 customer reviews)
  • Amazon Best Sellers Rank: #1,427,574 in Books (See Top 100 in Books)

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44 of 44 people found the following review helpful
5.0 out of 5 stars Deserves 10 stars September 21, 2002
Format:Hardcover
This book should have been titled "A Hilbert Space Idea/Problem Book" as it not only challenges the reader to work out interesting problems in operator theory and the geometry of Hilbert space, but also motivates the essential ideas behind these fields. It is definitely a book that, even though out-of-print, will be referred to by many newcomers to operator theory and quantum physics. The insight one gains by the reading of this book is unequaled in any other books in existence on operator theory. It is becoming more rare as mathematics advances, to find books that attempt to explain the intuition behind the abstractions that are manifested in any area of mathematics. The problems in the book deal with both concrete examples and general theorems, and the reader should attempt to try and solve them without looking at the hints. The solutions found by the reader can then be compared with the author's, and some interesting differences will occur.
There are so many interesting discussions in this book that to list them all would probably entail listing everything in the book. The reader will find excellent discussions of the origin of normal operators on infinite dimensional Hilbert spaces as analogs to matrices on finite dimensional spaces; why the weak topology in infinite dimensions is not metrizable; the non-emptiness of the spectrum and why the spectral radius can be computed even though the spectrum cannot; the impossibility of isolated singular operators; the non-continuity of the spectrum: the existence of an operator with a large spectrum and the existence of operators with small spectra in every neighborhood of the large spectrum.
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Format:Hardcover|Verified Purchase
The problems in this text are a little difficult, but this book is a great resource. I haven't done a complete reading of the text, but it had a very nice treatment of a problem that I needed for a research paper. Chances are, if you're faced with a Hilbert Space problem you can't figure out, it's broached in this text in some form.
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