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Prerequisites: 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. These prerequisites 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.

Ideally, students coming into this courses 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. All these ideas latter topics play roles in 8602.

This year, we're trying an experiment, to make 8602 depend less upon 8601, so we'll succinctly review necessary ideas about measure-and-integration.

Diagnostics: A brief diagnostic questionnaire is available, for self-evaluation, by prospective students, of their readiness for 860x. 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.

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

grades will be determined by four 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: briefly after class, and email anytime!

Sample examples for the course

• Overview/preview: extending/refining notions of "function", "integral", "derivative" and "limit"
• Metrics and topologies on vector spaces [ updated 30 Dec '19]
• 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.
  • Examples 01-02 ... Discussion 01 ... Discussion 02
• Riesz-Markov-Kakutani: a characterization 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]
• 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 03-04 ... Abbreviated Examples 03-04 ... Discussion 03 Discussion 04
  • Lemmas: partitions of unity, supports of distributions [ updated 18 Dec '22]
  • Intro to Sobolev spaces [ updated 05 Jan '20]
  • 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 05-06 ... Discussion 05 ... Discussion 06
  • 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]
  • 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
• 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 07-08 ... Discussion 07 Discussion 08