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Prerequisites for 8601: strong understanding of a year of undergrad real analysis, such as our 5615H-5616H or equivalent, with substantial experience writing proofs . Courses named Advanced Calculus are insufficient preparation. The necessary mathematical background includes careful treatment of limits (of course!), continuity, Riemann integration on Euclidean spaces, basic topology of Euclidean spaces, metric spaces, completeness, uniform continuity, pointwise limits, uniform limits, compactness, and similar.

General comfort with abstraction is a prerequisite.

Substantial experience writing proofs is a prerequisite. Ideally, students coming into this course have acquired a range of experience in proof writing, not only in a previous course in real analysis, but also in previous courses in abstract algebra, rigorous linear algebra, or point-set topology. For that matter, all the latter topics play roles in 8601-02.

Coherent writing, apart from proof-writing itself, is essential. Contrary to some myths, the symbols do not speak for themselves.

Diagnostics: A brief diagnostic questionnaire is available, for self-evaluation, by prospective students, of their readiness for 8601. The meanings of all the questions, and some ideas about the answers, should be very familiar to prospective students already. The conduct of the course will presume so. Also, within the first week or so of the actual course, we will have an in-class diagnostic midterm, a review of the prerequisites. Difficulty in that exam would be a sign that one is not prepared for the course. By that point, it would likely be very difficult to catch up, and it would surely be wise to switch to the 5xxx-level analysis course.

Note: even with a somewhat thin background, systematic study over the summer can greatly improve one's preparedness for this course. However, one should not under-estimate the level of effort required to emulate one or more serious year-long courses in abstract mathematics in a few months over the summer.

Prerequisites for 8602: 8601 or equivalent.

Fall 2022 - Spring 2023

Text will be notes posted here, supplemented by whatever books or other notes you like.

grades will be determined by in-class midterms , scheduled as in the table below. There is no final exam. You are not competing against other students in the course, and I will not post grade distributions. Rather, the grade regimes are roughly 90-100 = A, 75-90 = B, 65-75 = C, etc., with finer gradations of pluses and minuses. So it is possible that everyone gets a "A", or oppositely. That is, there are concrete goals, determined by what essentially all mathematicians need to know, and would be happy to know.

There will be regular homework assignments preparatory to exams, as scheduled below, on which I'll give you feedback about mathematical content and writing style. The homeworks will not contribute to the course grade, and in principle are optional, but it would probably be unwise not to do them and thus fail to get feedback. No late homeworks will be accepted, for logistical reasons. Homework should be typeset, presumably via (La)TeX, and emailed to me as a PDF.

Office hours: Mon 11:15-12:00, Wed 11:15-12:00. Send email anytime!

• Overview/preview: extending/refining notions of "function", "integral", "derivative" and "limit"
• Metrics and topologies on vector spaces [ updated 30 Dec '19]
  • Examples 01 [ updated 17 Sep '22] due Monday, Sept 26, 2022 ... Discussion 01 [ updated 27 Sep '19]
• Basic inequalities [ updated 20 Oct '19]: Cauchy-Schwarz-Bunyakowski, Young, Jensen, arithmetic-geometric mean, Holder, Minkowski
• Measure and integral [ updated 17 Oct '19]: a classical attempt at an extended notion of function: measurable functions. Integrating measurable functions. Fatou Lemma, Lebesgue Monotone Convergence, Lebesgue Dominated Convergence. Fubini-Tonelli. Abstract integration.
  • Discrete Fubini-Tonelli Illustrating monotone families with counting measures
  • Examples 02 [ updated 17 Sep '22] due Monday, Oct 17, 2022 ... Discussion 02 [ updated 24 Oct '19]
• Riesz-Markov-Kakutani, and a conceptual idea of "integration" [ updated 27 Oct '19]
• Example: L^p spaces [ updated 26 Oct '19]
• Hilbert spaces and a correct minimum principle (Dirichlet principle) [ updated 20 Oct '19]
• Fourier series [ updated 26 Oct '19] and Fourier transforms [ updated 25 Feb '20] convolutions [ updated 06 Jan '20]
  • Examples 03 [ updated 17 Sep '22] due Monday, Nov 07, 2022 ... Discussion 03 [ updated 19 Nov '19]
• Abstract Banach spaces [ updated 05 Jan '20] ... Applications of Banach space ideas [ updated 05 Jan '20]
• Generalized functions (distributions)
  • Intro to distributions (generalized functions) [ updated 16 Jan '20]
    • Examples 04 [ updated 17 Sep '22] Discussion 04 [ updated 14 Dec '19]
  • Lemmas: partitions of unity, supports of distributions [ updated 18 Dec '22]
  • Intro to Sobolev spaces [ updated 05 Jan '20]
    • handwritten 20 Jan 2023
  • Examples of Fourier transforms relevant to Sobolev spaces [ updated 31 Mar '20]
  • Smooth functions on [a,b] do not form a Banach space [ updated 05 Jan '20]
  • Extension by continuity [ updated 16 Jan '20]
  • (Generalized) functions on circles [ updated 05 Jan '20]
    • Examples 05 [ updated 17 Jan '23] 2020 ... Discussion 05 [ updated 12 Feb '20]
  • Fourier transform of hyperbolic secant [ updated 21 Mar '22]
  • Distributional proof of Poisson summation [ updated 13 Feb '23]
  • Paley-Wiener theorems [ updated 05 Jan '20]
  • Peano-Borel theorem on arbitrary Taylor-Maclaurin coefficients [ updated 02 Feb '20]
  • Iconic examples of distributions [ updated 28 Jun '21]
    • handwritten 22 Feb 2023
    • handwritten 27 Feb 2023
    • Examples 06 [ updated 17 Jan '23] 2020 ... Discussion 06 [ updated 04 Mar '20]
  • trace theorems [ updated 31 Mar '20]
• Introduction to spectral theory of operators on Hilbert spaces
  • Continuous (=bounded) linear operators [ updated 12 Mar '20]
  • Compact operators [ updated 12 Mar '20]
  • The family sin(nx/2) [ updated 28 Mar '20] via the spectral theorem for compact self-adjoint operators, and a simple Green's function
  • The family exp(inx) [ updated 30 Mar '20] via the spectral theorem for compact self-adjoint operators, and a variant Green's function
    • Examples 07 [ updated 01 Apr '23] ... Discussion 07 [ updated 28 Mar '20]
• Introduction to Schwartz' Kernel Theorem, traces, etc.
  • everywhere-defined, self-adjoint, imply continuous (=bounded) [ updated 12 Apr '23]
  • Schwartz kernel theorem, nuclear spaces [ updated 07 May '20]
    • Examples 08 [ updated 01 Apr '23] ... Discussion 08 [ updated 26 Apr '20]
• Bonus: Minkowski's theorem on convex bodies and generalization to topological groups [ updated 30 Apr '23]

Exam and homework-example schedule, spring 2023: