Análise no R^n - COVID19 edition

IMPA - Mar 3rd to Jul 3rd,2020

News

  • ----------------- SECOND EXAM IS HERE -------------------
  • [25/06] Try your hand at the "lista adicional" at the bottom of the page. Some of its problems may be in the exam.
  • Information about final exam is now available. See below.
  • [12/06] Class this week is on Friday at 10h30. See videos below!
  • -------------------- FIRST EXAM IS HERE ----------------------
  • [08/05] Our class day for that week (after the exam).
  • [29/04] The class on that day has been moved to April 30th at 10h30.
  • [22/04] Class postponed to the 23rd at 10h30. Also note that the chapter on compactness in the lecture notes has been updated.
  • [14/06] The lecture notes have been updated in the chapter on connectedness.
  • [13/06] Q & A session on Monday at 12h30.
  • [11/04] The new due date for the first four homework assignments is April 20th.
  • [09/04] Online question and answer section at 16h00.
  • [02/04] A new link to the lecture notes is here.
  • [25/03] This is a completely new version of the course webpage. The old version is here.

General information

  • Instructor: Roberto Imbuzeiro Oliveira
  • TA (monitora): Cynthia Bortolotto
  • Class format during COVID19 shutdown: lecture videos and handouts will be uploaded each week. There will also be weekly online meetings each Wednesday at 10h30 GMT-3. Please register for the class in order to be notified of these meetings.

About the class

This is a class on "Analysis beyond the real line"; not just "Analysis on R^n". This class will prepare you for mode advanced classes like Measure and Integration and Functional Analysis. We will cover the following topics.

  1. Topology and Analysis over metric and vector spaces (main examples: R^n and space of continuous functions).
  2. Fréchet derivatives and their properties.
  3. Inverse and Implicit Function Theorems. A bit about submanifolds of R^d.
  4. Classical theorems on the space of continuous functions (Ascoli-Arzèla, Stone-Weierstrass).
  5. Line integrals, multidimensional integrals.

Everybody is welcome to take the class. However, the class was designed for Masters students at IMPA, with a thorough knowledge of Analysis over the real line. The main prerequisites are: axiomatic definition of the real line; sequences and series; continuous and differentiable functions; and integration of continuous functions. Students are expected to exhibit a degree of mathematical maturity; in particular, they should be able to fill gaps and do exercises that will be essential for the class.

Grading

The following dates are subject to change. Students should assume we will have activities up until the official end of the term.

  • Initial homework assignments: students should submit solutions to the first four homework assignments by email to the TA by April 15th 20th. Those who do so will receive "participation points" as well as feedback on their solutions.
  • Second batch of homework assignments: HW 5 and 6 (to be posted) must be sent by email to the TA by May 6th. Your grade in these will be combined with the "participation points" above to replace what would be the first test.
  • A first exam will be posted on this webpage on May 6th at 9h, Rio de Janeiro time. Your solutions to the problems must be sent by email to me by 21h on May 7th (Rio de Janeiro time again). Even if we resume physical classes at IMPA, this first exam will be a take-home exam.
  • The second test consists of the seventh and eighth HW assignments, which must be turned in by email to the TA by June 5th. This test will be graded. .
  • The second and final exam will be posted on this webpage on July 2nd at 9h, Rio de Janeiro time. Your solutions to the problems must be sent by email to me by 21h on July 3rd (Rio de Janeiro time again).
  • The remaining HW assignments to be turned in will be due on July 2nd at 9h.
  • Final numerical grade = TBA
  • Letter grade will be A,B,C or F decided based on comparative evaluation of the letter grades.
  • In exceptional circumstances, requests for make-up/replacement exams will be considered. Be prepared to justify your request, and present evidence to corroborate your justification.

Bibliography

My lecture notes (in Portuguese - last updated on April 14th) are the main reference. You can also follow the course via the handouts and videos in English posted below.

Useful books.

Ralph Abraham, Jerrold Marsden & Tudor Ratiu. Manifolds, Tensor Analysis and Applications (Springer).

Rolci Cipolatti, Cálculo Avançado (SBM).

Serge Lang, Undergraduate Analysis (Springer).

Topics covered, with handouts and videos

Week of March 11th - no handout

Definitions of vector spaces, norms and metric spaces. Relationship between norms and linear functionals. Completeness of R^n. Pointwise versus uniform convergence for continuous functions.

Weeks of March 18th and 25th

Convergence and continuity. Roughly corresponds to Chapters 3 and 4 of my lecture notes.

  1. C([0,1],R) is complete with sup norm - https://youtu.be/WpTcXkbw7Ks
  2. Equivalent metrics and norms - https://youtu.be/w8L4JQurGpc
  3. Continuity: basics - https://youtu.be/Tu2MO0mUa38
  4. Lipschitz functions & functions related to distances - https://youtu.be/DK1X7LCQHf4
  5. Linear transformations - https://youtu.be/PY-EVkO9ckk
  6. Multilinear transformations - https://youtu.be/_ea2U2Ww70k
  7. One last video with examples (convolutions, ODEs) - https://youtu.be/B6zQp_x6Lsc

Week of April 1st

  1. Open and closed sets/balls - https://youtu.be/JffLeRfD8iw
  2. General topology: interior, closure, boundary, accumulation set &c - https://youtu.be/hf2Wx-zOY1A
  3. Metric characterizations of the above - https://youtu.be/_65apUol7eI
  4. Continuity & topology - https://youtu.be/CoCv1fGcpwc
  5. Induced topology and induced metric - https://youtu.be/xIP6WTq74UM

Week of April 8th

  1. Pathwise connectedness: definition, examples, properties - https://youtu.be/G4x4ia36U7E
  2. Topological connectedness: definition, examples, properties - https://youtu.be/Y46TS0xJBPA
  3. Pathwise <=> topological for open sets in vector spaces - https://youtu.be/oPI0IjtsWco
  4. An example showing that pathwise connectedness does not imply topological connectedness - there will be no video of this, as there are many proofs written down elsewhere. See e.g. http://math.stanford.edu/~conrad/diffgeomPage/handouts/sinecurve.pdf.

Weeks of April 15th and 22nd

  1. Compact spaces: topology, boundedness of continuous functions - https://youtu.be/Qi92KicvvOI
  2. Compact metric spaces: "grand equivalence theorem", all compact are complete - https://youtu.be/2chf49OTA_Y
  3. Totally bounded sets: definitions, compact => totally bounded - https://youtu.be/lVOmwNRJ51Q
  4. Complete and totally bounded => all sequences have convergent subsequences - https://youtu.be/Am5KkJzlVFs
  5. All sequences have convergent subsequences => topological compactness (end of grand equivalence) - https://youtu.be/PV4jw4f531c
  6. Compact subsets of metric spaces. The case of R^d: Heine Borel and the equivalence of all norms - https://youtu.be/pN4z3sNAQ8w
  7. Continuous functions with compact domain - basic properties (incl. uniform continuity) - https://youtu.be/8zWf-YzjRFA
  8. Ascoli-Arzèla: a criterion for compactness of sets of continuous functions. - https://youtu.be/J3gZsx0A1fc

Week of May 13th

  1. Basic differential calculus for f:R -> vectors, including Mean Value Inequality - https://youtu.be/0m47H0huwSI
  2. Integrals and the Fundamental Theorem of Calculus for f as above - https://youtu.be/C_-S_xG8DKA
  3. Convergence of series and sequences of functions. Applications to power series. - https://youtu.be/qwrHAaNVCcA

Week of May 20th

  1. Fréchet derivatives of functions f: vectors-> vectors: motivation, relation to 2, chain rule. - https://youtu.be/Hi9hFzXH0V8
  2. Fréchet derivatives for R^d->R^k. Relationship to partial derivatives and Jacobian matrices. - https://youtu.be/_zp9exSS4XI
  3. Mean Value Inequality. Fréchet derivatives in infinite dimension: some examples. - https://youtu.be/hZYWdFZU2y8

Weeks of May 28th and June 3rd

1. Second Fréchet derivatives - https://youtu.be/p1TpUotvqgs

2. Symmetry of second derivative (when continuous) - https://youtu.be/JRrYPxs929E

3. Third and higher derivatives - https://youtu.be/7nXzBI5oFLE

4. Taylor series - https://youtu.be/LzEtGfj1oMc

Week of June 12th and 19th

  1. Banach's Fixed Point Theorem and Application to ODEs - https://youtu.be/AZ__ZY5IG8Q
  2. The Inverse Function Theorem, Part 1 - https://youtu.be/aQ9oxoGLw6s
  3. The Inverse Function Theorem, Part 2 - https://youtu.be/Rs1smbGIHhs
  4. The Implicit Function Theorem - https://youtu.be/c7Hdl-s0Rdg

Week of June 24th

  1. Introduction to submanifolds of R^d: definitions and the tangent space - https://youtu.be/9_L58C9UWKs
  2. Implicitly defined submanifolds and a bit of Optimization - https://youtu.be/8nH1oAXFdYU
  3. Some interesting examples (TBA)