Math 5210-1
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Lectures will be posted here!
Week 1: Jan 12-16
jan12.pdf measuring distance
jan13.pdf 2.1 equivalence relations, Cauchy sequences, and the real numbers.
jan14.pdf 2.2 The real numbers as an ordered field.
jan16.pdf 2.2-2.3 and the completeness of the reals.
Week 2: Jan 20-23
jan20.pdf 2.3 Limit theorems and subdivision arguments.
jan21.pdf 2.3 sups, infs, limsups, liminfs, lims.
jan21extra.pdf addendum
jan23.pdf 9.1 metric spaces
Week 3: Jan 26-30
Monday Jan 26: Use last Friday's notes!
jan27.pdf 9.1, 9.2: complete and incomplete metric space examples, and metric space completion.
jan28.pdf 9.2: open and closed sets, interior, closure, boundary.
jan28extra.pdf the complete notes on metric space completion.
jan30.pdf metric subspaces
Week 4: Feb 2-6
feb2.pdf 9.2: compactness
feb3.pdf 9.2: cont'd
feb4.pdf 9.2: equivalent metric space notions of compactness
feb6.pdf 9.3: continuous functions
Week 5: Feb 9-13
feb9.pdf 9.3: continuous functions on compact domains
feb10.pdf 9.3: connected sets; also the contraction mapping theorem.
feb11.pdf 9.3: Hausdorff distance between compact sets, constructing fractals as an application of the contraction mapping theorem.
ClassicalFractals.pdf
IFSfractals.pdf
fractal directory
feb13.pdf 9.3: the google page rank algorithm is an application of the contraction mapping theorem.
Week 6: Feb 17-20
feb17.pdf 11 prep: The Riemann integral for functions with Banach Space range
feb18.pdf 11 prep: Differentiability for maps between Banach spaces. (See also section 10.1 of text.)
feb20.pdf 11 prep: comments on previous notes, and new hw.
Week 7: Feb 22-26
feb23.pdf 11.1-11.4: Contraction mapping proof for local existence and uniqueness to the IVP, for first order systems of DEs... plus two more class exercises.
feb24.pdf 11.1-11.4: Extending the local existence-uniqueness theorem to global existence-uniqueness theorems.
feb25.pdf 11.1-11.4: Matrix exponential solutions example; interchanging limits and integrals.
feb27.pdf 7.3: interchanging limits and derivatives with Riemann integrals.
Week 8: Mar 2-6
mar2.pdf 7.4 Power series; term by term integration and differentiation.
mar3.pdf 7.5 Convolution and the Weierstrass Approximation Theorem.
mar4.pdf 9.3 The Stone-Weierstrass theorem generalizes the Weierstrass Approximation theorem.
mar6.pdf exam review sheet, and solving the heat equation with our particular approximate identity.
Week 9: Mar 9-13
mar9.pdf interchanging derivatives and integrals
mar10.pdf 7.6, 9.3: compact subsets of C(X,Y) when X and Y are compact metric spaces, i.e. the Arzela-Ascoli Theorem.
mar13.pdf midterm results and next hw assignment
Week 10: Mar 23-27
mar23.pdf 12: introduction to Fourier series
mar24.pdf 12.2-12.3 uniform and pointwise convergence theorems for Fourier series.
mar25.pdf complex vs. real inner product spaces
mar27.pdf Chapter 14 Strichartz, Chapter 3 Royden: introduction to measure theory
Week 11: Mar 30 - Apr 3
mar30.pdf Chapter 3 Royden: outer length measure and measurable sets
mar31.pdf Chapter 3 Royden: measurable sets and Borel sets.
apr1.pdf Chapter 3 Royden: measurable sets and Borel sets, cont'd.
apr3.pdf Chapter 3 Royden: the algebra of measuring measurable sets, and the Cantor set/function example.
Week 12: Apr 6 - 10
apr6.pdf transition to Strichartz 14.2, metric outer measures.
apr7.pdf 14.2, metric outer measures and Hausdorff measure/dimension.
apr8.pdf 14.2: metric outer measures yield measures on a sigma algebra containing the Borel sets.
apr10.pdf intro to measurable functions
Week 13: Apr 13 - 17
apr13.pdf Measurable functions (Royden 3.5)
apr14.pdf Royden 3.6, 3.4
apr15.pdf Royden 4.1-4.2; simple functions and Lebesgue integration.
apr17.pdf Royden 4.2 continued.
Week 14: Apr 20-24
apr20.pdf integral properties and the bounded convergence theorem, 4.2.
apr21.pdf Monday continued, and introduction to 4.3: the Lebesgue integral for non-negative measurable functions on arbitrary measurable domains.
apr22.pdf 4.3-4.4 The Lebesgue integral for non-negative measurable functions on arbitrary measurable domains, and the general Lebesgue integral.
apr24.pdf 4.4 continuous versions of the limit theorems; series as Lebesgue integrals.
Week 15: Apr 27-29
apr27.pdf some final exam problems (optional take home), and Royden's non-measurable set.
apr28.pdf L^p spaces, and the rest of the final exam mitigation problems.
apr29.pdf course topic review sheet