- Series: Applied Mathematical Sciences (Book 40)
- Paperback: 624 pages
- Publisher: Springer; 1st ed. 1982. 2nd printing 2000 edition (February 23, 2000)
- Language: English
- ISBN-10: 038795001X
- ISBN-13: 978-0387950013
- Product Dimensions: 6.1 x 1.5 x 9.2 inches
- Shipping Weight: 2 pounds (View shipping rates and policies)
- Average Customer Review: 5.0 out of 5 stars See all reviews (6 customer reviews)
- Amazon Best Sellers Rank: #1,602,732 in Books (See Top 100 in Books)
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Linear Operator Theory in Engineering and Science (Applied Mathematical Sciences) 1st ed. 1982. 2nd printing 2000 Edition
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"Vivid and easily understandable . . . numerous exercises . . . have greatly advanced the understanding of material presented in the book . . . can be used by students with various level of preparation with different interests, as well as a textbook for a senior-level courses."―ZENTRALBLATT MATH
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Top customer reviews
How best to study this text ? Start with the Appendix A: Schwartz, Holder, Minkowski Inequalities.
Memorize them--they will be utilized throughout the text. Onward: The first two chapters will be in the
nature of review. What is reviewed ? Sets, Functions, Matrices, Modeling, especially, Proof by Induction.
Attempt to solve each problem in the first two chapters. Few are difficult, all will be relevant in the remainder.
Hints are provided for the solutions to many problems in the text. While the definition-theorem-proof format
is generally evidenced, it is the copious number of examples which are distinctive. Trilogy of pedagogic strategy
is made manifest: Algebra, Geometry, Topology. Trilogy of possible pathways is employed : Parts A, B, & C.
At a minimum, all of Part A should be assimilated. Then, the more detailed, and technical, B's and C's can be
picked and studied at one's leisure. Minkowski Inequality for sums and integrals is used, third chapter (Page 49 & 51).
A nice--and easy--problem occurs Page 61, #3, which reinforces the concept of pseudometric. Schwartz Inequality
for integrals is used (Page 65). Holder and Lipschitz conditions presented in Problems 7 & 8 (Page 68). Section 3.7
gives a nice discussion of the connection between continuity and convergence. Scattered throughout are phrases
which begin "...intuitively speaking..." or "...let us recall the concept of..." or "...an analogous situation arises...,"
these are segues into the more physical applications of the abstract mathematics. The Example 5 ( Page 179), of
interest, is illustrative of a linear subspace being spanned by a set. Convexity is introduced (Page 182) via Problem 11.
Section 4.8, elucidation of matrices to represent linear transformations, is very clear, indeed.
Pauli Spin Matrices introduced in the problem set which ends this Part A of fourth chapter.
Fifth chapter: combining Topology and Algebra. Banach and Hilbert make their appearances.
Sequences and Infinite Series, "the first offspring of this wedding of topological and algebraic structure" (Page 224),
the most "important offspring" is the concept of continuous linear transformation" (Page 234). Cauchy's Integral Formula
for analytic functions is utilized here. "We urge the reader to master them--Theorems 5.6.2 & 5.6.4--before continuing."
Fourier Series Theory will be met in short order. Gram-Schmidt Process will be met subsequently.
An easy-to-follow example (#3-- Page 316) will add understanding to the concept of orthonormal sets in Hilbert Space.
Zorn's Lemma (Axiom of Choice, Appendix B) is used often throughout. Integration and Measure Theory are reviewed.
Example 4 (Page 326) Multiple Fourier Series, is another easy-to-follow derivation. Following which are examples with
Hermite and Laguerre Functions. Riesz Representation Theorem is nicely elucidated, Problem Sets which exemplify
such, end section 5.21--quite an interesting potpourri (Pages 345-351). Chapter Five, Part C, Special Operators,
is a highlight of the exposition. Fourier Transforms exemplified with lucidity.
Nice Problem Sets, also: Look at Page 376, #7,or #24, both eminently do-able( Concrete and Abstract in tandem).
Quantum Mechanics gets a page--or two--we read: "a state is defined as a probability function defined on the collection
of yes/no experiments." Reference (for details) made to the classic text of Jauch (1968, Foundations of Quantum Mechanics).
The final two chapters: Analysis of compact, then Unbounded, Linear Operators. We meet Eigenvalues and Spectrums, first.
Second, we meet Green's Functions. A nice flow chart (Page 413) "...may aid the reader in remembering the contents."
Examples of spectra abound. Study all of the examples in this section 6.6 (there are eight, few details are omitted).
The Spectral Theorem and its Applications finish this most satisfying Chapter. The last chapter is exceptional:
Unbounded Operators. And, again, approachable exercise sets (example: Page 492, Number 3, four sequential parts).
Laplace, Dirichlet, Elliptic and Wave Operators all make acquaintance. Again, Schwarz Inequality effectively utilized.
And, as an added bonus, Quantum Mechanics is revisited (Heisenberg and Harmonic Oscillators given quick fly-by).
This ends the text. As one ascertains, there is an incredible amount of information between these covers.
The presentation is pedagogically masterful : Many examples, much reviewed, illustrations summarize much material.
An exceptional offering of introductory and advanced material.
The book is well organized, and it provides detailed discussion on each topic followed with numerous and easy to understand examples that are directly related to physical systems. The reader can visualize the concepts by relating to physical systems, which is what engineering students want. After four months of evening reading, it is amazing that now I can tackle all analysis problems that used to intimidate me. This book is a must for all graduate students whose major covers dynamical systems, controls, nonlinear systems or signal processing.