2016 summer bootcamp in analysis.

Time and place: MTWTh 10:00am to 11:50am, MS 5138.

Instructor: Itay Neeman.
Office: MS 6334.
Phone: 794-5317.
Office hours: TW 1-1:50pm, or by appointment.

Discussion session: Fridays 10 to 11:50am with Adam Azzam.

The summer bootcamp helps entering graduate students prepare for the Basic Qualifying Exam. The exam covers material in analysis and linear algebra. In this part of the bootcamp we work on the topics in analysis.

Lectures and homework in the class will cover a sampling of key results in topics from the exam syllabus, and many questions from past exams. But we will not be able to cover all the material for the exam, and so this course is not a substitute for studying on your own. You can find the full syllabus for the exam, and suggested textbooks which cover it, on the exam webpage.

There is no grading for the course, but there will be homework assigned each week. You should write down solutions to the assigned homework, and view the assignments as practice for the Basic Exam. If you have any questions about the assignments the TA and I can help during office hours. You should write your solutions down each week before Friday. The homework will not be graded or marked, but the TA will present solutions to some of the exercises and you can use the discussion on Friday to check that your solutions are correct.

The homework assignments are posted below. Many of the questions are taken from old exams. These are listed as XYY.N, where X indicates the term of the exam (Fall, Winter, or Spring), YY indicates the year, and N is the question number (taken from the analysis part of the exam in the rare case that the linear algebra and analysis parts are numbered separately). The exams are posted online here. Some homework questions will come from the books Analysis I and II by Tao, 3rd edition. These are listed as B.C.S.E where B is the book (either I or II), C is the chapter number, S is the section number, and E is the exercise number.

Homework assignments: (Updated as the term progresses. Exercises each week may rely on material from the same week and any previous week.)

Week 1. Material on cardinality, the real line, completeness, topology, connectedness, compactness, metric spaces, sequences, and convergence. Exercises II.2.5.6 and II.2.5.7 (in older editions, "Hausdorff" in both should be "not Hausdorff"), S03.3, F04.4, F04.6, S04.4, F05.8 (with \(\Vert A\Vert=\sup_{\vec{x}\in{\mathbb R}^n, \Vert\vec{x}\Vert=1} \Vert T_A\vec{x}\Vert\)), S05.8, W06.1, F07.1 (keep in mind \(S\) is an arbitrary set), S07.12, F08.4, S08.7 (on last line \(f(x)\) should be \(a(x)\)), F09.1, S09.4, F11.3, S12.1, S12.3, S12.4, S13.11.

Week 2. Material on convergence of sums, rearrangements and absolute convergence, continuity in topological and metric spaces, path connectedness, intermediate value theorem, contraction maps and the fixed point theorem, uniform continuity, uniform convergence, and the Arzela-Ascoli theorem. Exercises F04.5, S04.6, S05.7, S06.5, S07.10, F08.1, F08.2, S08.6, F09.6, S09.6, F10.4(b), F11.4, S11.7, F12.3, S12.2 (hint), S12.5 (hint), F13.2, F13.3, F13.6, S13.3.

Week 3. Material on definition of derivative, derivative for inverse function, local maxima and minima, Rolle's theorem, mean value theorem, Rolle's theorem for higher order derivatives and applications to error bounding for approximations by Lagrange interpolations, monotonicity, Cauchy mean value theorem, L'Hopital's rule, uniform convergence limits of derivatives (in homework), upper and lower Riemann integrals, upper and lower Riemann sums, definition of the Riemann integral, integrability of bounded continuous functions on bounded intervals, basic properties of the Riemann integral, integrability of mins, maxes, sums, and products, mean value theorem for integrals, Riemann-Stieltjes integral, the fundamenal theorems of calculus, integration by parts, change of variables in integration, improper integrals, integrals of uniform convergence limits (in homework). Exercises I.7.5.2, I.7.5.3, II.3.7.2, F04.3, F05.3, F05.4, S05.3, S05.4, S05.6, S06.3, S06.6, F07.2, F07.11, F08.10, S08.2, F09.2(b), F09.3, S10.12, S11.8, S13.1.

Week 4. Material on Cauchy-Schwarz, Young's, Hölder's, and Minkowski's inequaities, formal power series, radius of convergence, real analytic functions, absolute and uniform convergence on closed subintervals, derivatives and integrals of power series, Taylor's forumla, Abel's lemma, Abel's theorem for uniform convergence and continuity, Stone-Weierstrass theorem, Taylor theorem with reminder in Lagrange, Cauchy, and integral forms, Newton's methods for finding roots of a single function, error bounds in numerical integration and differentiation (homework), Fubini theorem for sequences, multiplication of power series, the exponential and logarithm functions. Exercises I.11.9.1 (for older editions, the function is \(F(x)=\int_{0,x} f\) and you should ignore the hint), I.11.9.3, W06.3 (\(1\over n\) on left should be \(b-a\over n\)), F07.4, F07.5, F07.9 (in part (b), assume \(u_n\) converge uniformly, and prove they converge uniformly to \(u\)), S07.6, S07.8, S08.3, S09.10, F10.4(a), F10.10 (hint), S11.9, S11.12, F12.1, S13.2.

Week 5. Material on the sine, and cosine functions, uniform approximation of periodic functions by trigonometric polynomials, multi-variable differentiation, the chain rule, partial derivatives, directional derivatives, differentiability of functions with continuous partial derivatives, inverse function theorem, implicit function theorem, Lagrange multipliers, integrals in several variables, change of variables, differentiation under the integral sign, integration over product of spaces and double integrals, Clairaut's theorem on equality of mixed partial derivatives, higher order directional derivatives, Taylor's formula for functions of several variables, connection to Newton's method in several variables, local minima, maxima, and saddle points in several variables, line integrals, Green's theorem, outline of Stokes theorem in \({\mathbb R}^3\), outline of divergence theorem in \({\mathbb R}^2\) and \({\mathbb R}^3\). Exercises II.4.1.2, II.4.5.8, II.4.5.4, II.6.7.3, F03.6 (hint), F04.7, F04.8, S04.5, W06.5, S07.11, F09.8, S09.12, F10.3, F10.12, S10.9, S10.10, S12.6, F13.5, S13.4, S13.5 (subscripts \(1\) and \(2\) should be \(0\) and \(1\) respectively).

Good luck on the Basic Exam!