Linear Operator Theory in Engineering and Science by A.W. Naylor,G.R. Sell

I'm doing a PhD in econometrics and I need to apply operator theories in constructing a linear or nonlinear operator to help explain individual economic behaviour. This book contains numerous useful ideas and applications with exercises thoroughly designed; one of the questions in the exercise gave me an idea of creating a matrix for describing a nonlinear operator. That question asks for a matrix that describes a second order differential operator and that gave me an idea that taylor series approximation can be used to linearise a nonlinear operator and hence a nonlinear operator may also be described by a matrix.

Very Nice Book!
So many examples, almost all the theorems having proofs.
Many graphs, which help comprehension a lot!

If searching for a unique and approachable exposition of Linear Operators, your search ends here.
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.
Highly Recommended.

At graduate school, I was required to take only 9 credit units in advanced Mathematics. Most of my credits were on Numerical Analysis, Mathematical Optimization and Linear Algebra. However, as I started covering advanced courses in dynamical systems, robust control and nonlinear systems, I found that I really needed strong foundations in Functional Analysis. I audited one course in Functional Analysis where we used the classic Rudin text. However, the pace of that class was too fast for me and I virtually learnt nothing. So I graduated with a handicap that I was unable to handle most rigorous problems. Knowing my handicap, I bought several books on this subject; I did not want to spend too much money, so each book did not cost more than $50 on Amazon marketplace. At first it was very disappointing because each book that I bought was written in the traditional format of Definition-Theorem-Proof without explaining the basic concepts. The last book to buy was this one which I read chapter by chapter from page one to the last page for about four months every evening. I believe this will be my last book on functional analysis, because it has provided what I have been looking for.

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.

This is one of my favourite introductory books on functional analysis because it includes all the fundamental concepts needed to get a thorough understanding of the most important topics in this theory. It stresses which properties come from which structure (set-theoretic, topological, or algebraic), and gives a complete exposition of the main results on bounded and unbounded operators, meanwhile discussing some interesting examples of application. Its appendices on integration, probability and stochastic processes are also valuable. The text is suitable for undergraduate analysis and functional analysis courses. Scientists and engineers will find it very useful also because of its non-pedantic style.
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