Functional Analysis

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1. graycatJul 2025
   "Functions Are Vectors"?

   Let's see: Let A be a non-empty set, N the set of positive whole numbers, and X the set of all functions f

   f: A --> N

   with usual notation.

   Assume as is common, the scalars are the set of real numbers, but the set of complex numbers will also do.

   So, is X a vector space and, thus, each f in X a vector?

   No, since -f is not in X. Neither is (1/2)f.

   Some references (with TeX markup):

   Paul R.\ Halmos, {\it Finite-Dimensional Vector Spaces, Second Edition\/}

   linear algebra treated as functional analysis.

   Walter Rudin, {\it Real and Complex Analysis\/}

   with Lebesgue integration and, then, Banach and Hilbert vector spaces.

   Walter Rudin, {\it Functional Analysis\/}

   with Fourier theory.

   Jacques Neveu, {\it Mathematical Foundations of the Calculus of Probability\/}

   with random variables, that is, functions from a probability space to, usually, the set of real numbers with convergence results, building on the work A. Kolmogorov building on the work of H. Lebesgue.

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2. eyphkaMay 2022
   Books that are given nicknames by grad students. :) For a foundational understanding of Calculus, and just becoming a better mathematician, a good start would be ->

   Baby Rudin (Principles of Mathematical Analysis, Walter Rudin)

   Papa Rudin (Real and Complex Analysis, Walter Rudin)

   Grandpa Rudin (Functional Analysis, Walter Rudin)

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3. graycatOct 2019
   > Of the books that I explored, Baby Rudin came the closest to this ideal. While many of the individual chapters contains long runs of mostly linear development, the interdependence of the chapters is complex, and apparently very carefully considered.

   Naw! Take another pass through Baby Rudin!

   Here is some help: With appropriate mild assumptions, a continuous, real value function on a compact set is uniformly continuous. For the finite dimensional space of interest, compact is the same as closed and bounded. That is the main content of the first chapters.

   First Application: With uniform continuity, get to show that the Riemann and Riemann Stieltjes sums converge so that those integrals do exist.

   Second Application: The uniform limit of a sequence of continuous functions is continuous. This was a question on my Analysis Ph.D. qualifying exam -- I got it! Thus a certain normed space is complete, that is, is a Banach space -- I'm not sure this remark is in the book.

   The chapter(s) on sequences and series is used to define ln, sin, cosine so that can do Fourier series.

   The exterior algebra is essentially separate.

   That's the main stuff.

   Don't take the Riemann integral very seriously: The good stuff is Lebesgue's integral (partitions the range instead of the domain) as in the first half of Rudin's Real and Complex Analysis. There also get to see Banach space, Hilbert space, von Neumann's novel proof of the Radon-Nikodym theorem (grown up version of the fundamental theorem of calculus and with astounding power) and the Fourier integral. I won't take off points if you don't read the second half.

   Rudin's Functional Analysis -- get to see distributions.

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4. graycatJan 2019
   Part III

   Rudin has a nice chapter on Fourier series, that is, representing a periodic function f(t) with a sum of e^(iwt) for overtones at frequencies w and where each frequency w is a whole number multiple of a fundamental frequency. The linear algebra and geometry here are that the e^(iwt) are perpendicular projections of the f(t). So, the e^(iwt) are orthogonal axes.

   The line integral generalizes to the exterior calculus of differential forms (keep track of signs from the direction do the integrations, i.e., as in line integrals) and the fundamental theorem of calculus generalizes to Stokes theorem crucial in Maxwell's equations, fluid flow, and partial differential equations. Now we are close to differential geometry, and Rudin also does the inverse and implicit function theorems important in differential geometry and parts of nonlinear optimization, e.g., Lagrange multipliers.

   I warmly suggest that take a really fun, one weekend, pass through Stokes theorem parts of

   Tom M. Apostol, Mathematical Analysis: A Modern Approach to Advanced Calculus, Addison-Wesley, Reading, Massachusetts, 1957.

   Get it used, and pay whatever you have to. This way get to see the 2-3 dimensional versions and, really, mostly enough for what physics and engineering do with vector analysis and Stokes theorem.

   In H. Royden, Real Analysis and the first half of W. Rudin, Real and Complex Analysis, can see the Lebesgue integral, due to H. Lebesgue near 1900. In essentially all cases where the Riemann integral is defined, the Lebesgue integral is also defined and gives the same numerical value. But the Lebesgue integral has more powerful theorems and is defined in more general situations.

   As in a 1933 paper of A. Kolmogorov, the Lebesgue integral gives a solid foundation to probability, statistics, and stochastic processes.

   In W. Rudin, Functional Analysis get a treatment of distributions that cleans up what physics tries to do with the Dirac delta function and also covers spectral theory.

   From there, can go for a Ph.D. For that will need mostly (A) pass the qualifying exams and (B) do some original research. The standards are commonly something like "an original contribution to knowledge worthy of publication" and for publication, "new, correct, and significant".

   For the qualifying exams, first pick a department, pure/applied math or some math area in engineering, optimization, probability, economics, computer science, etc., get their description of their qualifying exam topics and references, study, and take and pass the exams. For the research, do some.

   One suggestion: For the research, pick a real problem and use some math, at least in part new, to get a good solution. Can get "significant" from the importance of the real problem. Can get much of "new" if have the first or better attack on the real problem. Can get "correct" if do the math with careful theorems and proofs. Can publish if pick an appropriate journal. Might get to regard the math as significant based on what it does for the real problem and not just its contribution to pure math.

   Even the best US grad schools are hungry for good students. With a good ugrad math major and good work on independent study, should get a good reception at grad schools and at least a tuition scholarship. In that way I got accepted to grad school at math departments at Cornell, Brown, Princeton, and more.

   In a sense, independent study is recommended: (A) That is basically what research profs have to do for all their careers. (B) At least at one time the math department at Princeton stated that no courses were offered for preparation for the qualifying exams, students were expected to prepare for the exams on their own, courses were introductions to research by experts in their fields, and students should have some research underway in their first year.

   For a little on how to do the original parts of math research, there is the A. Wiles comment:

   "Perhaps I could best describe my experience of doing mathematics in terms of entering a dark mansion. You go into the first room and it's dark, completely dark. You stumble around, bumping into the furniture. Gradually, you learn where each piece of furniture is. And finally, after six months or so, you find the light switch and turn it on. Suddenly it's all illuminated and you can see exactly where you were. Then you go into the next dark room ...."

   That's some of how to learn some math, largely with independent study, and maybe to get a Ph.D. But you might want more: You might want your Ph.D. work to get you a good start on a good career in pure/applied math research/applications. For that, look around, pick up what you can in seminars and conferences, and get what your profs can explain to you.

   Broad point: Long the best opportunities in applied math, even with some advanced pure math prerequisites, have been in US national security, especially within 100 miles of the Washington Monument.

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5. graycatNov 2018
   Evar D. Nering, Linear Algebra and Matrix Linear Algebra and Matrix Theory*

   Kenneth Hoffmann And Ray Kunze. Linear Algebra, 2nd Edition, Prentice-Hall, Englewood Cliffs, New Jersey, 1971.

   https://www.zuj.edu.jo/download/linear-algebra-2nd-edition-k...

   Halmos, Finite Dimensional Vector Spaces

   George E. Forsythe and Cleve B. Moler, Computer Solution of Linear Algebraic Systems

   Paul R. Halmos, Naive Set Theory, Van Nostrand, Princeton, NJ, 1960.

   More has been done since this book, but this book is a gorgeous introduction to axiomatic set theory. So, even people who want to dig into the latest work would do well to have this as the first book. And for people wanting to read any of the more advanced material here, knowledge of this book will be from good to have to important.

   https://piazza-resources.s3.amazonaws.com/is25bi6c6o1oh/isv1...

   Walter Rudin, Principles of Mathematical Analysis

   The third edition is a lot better than the first two.

   H. L. Royden, Real Analysis: Second Edition

   Beautifully written, elegant, but maybe don't work way too hard on the exercises about upper/lower semi-continuity, and there is a better summary than Littlewood's three principles.

   Bernard R. Gelbaum and John M. H. Olmsted, Counterexamples in Analysis

   John C. Oxtoby, Measure and Category: A Survey of the Analogies between Topological and Measure Spaces

   Walter Rudin, Real and Complex Analysis

   Walter Rudin, Functional Analysis

   Leo Breiman, Probability

   Kai Lai Chung, A Course in Probability Theory, Second Edition

   Jacques Neveu, Mathematical Foundations of the Calculus of Probability

   Erhan Cinlar, Introduction to Stochastic Processes

   J. L. Doob, Stochastic Processes

   I. I. Gihman and A. V. Skorohod, The Theory of Stochastic Processes I, II

   Donald E. Knuth, The TeX book

   Donald E. Knuth, The Art of Computer Programming, Second Edition

   Leo Breiman, "Statistical Modeling: The Two Cultures," Statistical Science, Vol. 16, No. 3, 199–231, 2001.

   Paul R. Halmos, "The Theory of Unbiased Estimation", Annals of Mathematical Statistics, Volume 17, Number 1, pages 34-43, 1946.

   Paul R. Halmos and L. J. Savage, "Application of the Radon-Nikodym Theorem to the Theory of Sufficient Statistics", The Annals of Mathematical Statistics, Volume 20, Number 2 (1949), 225-241.

   https://projecteuclid.org/download/pdf_1/euclid.aoms/1177730...

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6. graycatNov 2018
   FDVS was written in 1942 when Halmos had just gotten his Ph.D. from J. Doob, author of Stochastic Processes in my list, at U. Illinois, and was an assistant to John von Neumann at the Institute of Advanced Study in Princeton.

   IIRC Hilbert space was a von Neumann idea: It is first, just a definition -- complete inner product (dot product in much of physics and engineering) space. But the good stuff is (1) importance of the examples and (2) the theorems that show the consequences, e.g., in Fourier theory.

   Well, the vector spaces of most interest in linear algebra are actually (don't tell anyone) finite dimensional Hilbert spaces. So, one role of FDVS is to provide a text on linear algebra that is also an introduction to Hilbert space, that is, that tries to use ideas that work in any Hilbert space to get the basic results in linear algebra.

   The treatment of self-adjoint transformations and spectral theory are likely the most influenced by this role.

   This role is accomplished so well that sometimes physics students starting on quantum mechanics are advised to get at least the start they need on Hilbert space from FDVS.

   Sure, a better start is the one chapter on Hilbert space in Rudin's Real and Complex Analysis. The chapter there on the Fourier transform is also good, short, all theorems nicely proved, the main, early results made clear.

   Also a good start on the basic results of self-adjoint matrices are the inverse and implicit function theorems given as nice exercises in the third edition of Rudin's Principles .... And spectral theory is in Rudin's Functional Analysis. Also get a bonus of a nice treatment of distributions, that is, replace the Dirac delta function usage in quantum mechanics.

   For how to get the eigen value and orthogonal eigen vector results for self-adjoint matrices from the inverse and implicit (these two go together like ice cream and cake) function theorems is in Fleming, Functions of Several Variables. Then you will be off and running on factor analysis, principle components, the polar decomposition, the singular value decomposition, and more.

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