----------------- 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.
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.
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.
Topology and Analysis over metric and vector spaces (main examples: R^n and space of continuous functions).
Fréchet derivatives and their properties.
Inverse and Implicit Function Theorems. A bit about submanifolds of R^d.
Classical theorems on the space of continuous functions (Ascoli-Arzèla, Stone-Weierstrass).
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.
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.
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).
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.
Convergence and continuity. Roughly corresponds to Chapters 3 and 4 of my lecture notes.
C([0,1],R) is complete with sup norm - https://youtu.be/WpTcXkbw7Ks
Equivalent metrics and norms - https://youtu.be/w8L4JQurGpc
Continuity: basics - https://youtu.be/Tu2MO0mUa38
Lipschitz functions & functions related to distances - https://youtu.be/DK1X7LCQHf4
Linear transformations - https://youtu.be/PY-EVkO9ckk
Multilinear transformations - https://youtu.be/_ea2U2Ww70k
One last video with examples (convolutions, ODEs) - https://youtu.be/B6zQp_x6Lsc
Open and closed sets/balls - https://youtu.be/JffLeRfD8iw
General topology: interior, closure, boundary, accumulation set &c - https://youtu.be/hf2Wx-zOY1A
Metric characterizations of the above - https://youtu.be/_65apUol7eI
Continuity & topology - https://youtu.be/CoCv1fGcpwc
Induced topology and induced metric - https://youtu.be/xIP6WTq74UM
Pathwise connectedness: definition, examples, properties - https://youtu.be/G4x4ia36U7E
Topological connectedness: definition, examples, properties - https://youtu.be/Y46TS0xJBPA
Pathwise <=> topological for open sets in vector spaces - https://youtu.be/oPI0IjtsWco
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.
Compact spaces: topology, boundedness of continuous functions - https://youtu.be/Qi92KicvvOI
Compact metric spaces: "grand equivalence theorem", all compact are complete - https://youtu.be/2chf49OTA_Y
Totally bounded sets: definitions, compact => totally bounded - https://youtu.be/lVOmwNRJ51Q
Complete and totally bounded => all sequences have convergent subsequences - https://youtu.be/Am5KkJzlVFs
All sequences have convergent subsequences => topological compactness (end of grand equivalence) - https://youtu.be/PV4jw4f531c
Compact subsets of metric spaces. The case of R^d: Heine Borel and the equivalence of all norms - https://youtu.be/pN4z3sNAQ8w
Continuous functions with compact domain - basic properties (incl. uniform continuity) - https://youtu.be/8zWf-YzjRFA
Ascoli-Arzèla: a criterion for compactness of sets of continuous functions. - https://youtu.be/J3gZsx0A1fc
Basic differential calculus for f:R -> vectors, including Mean Value Inequality - https://youtu.be/0m47H0huwSI
Integrals and the Fundamental Theorem of Calculus for f as above - https://youtu.be/C_-S_xG8DKA
Convergence of series and sequences of functions. Applications to power series. - https://youtu.be/qwrHAaNVCcA
Fréchet derivatives of functions f: vectors-> vectors: motivation, relation to 2, chain rule. - https://youtu.be/Hi9hFzXH0V8
Fréchet derivatives for R^d->R^k. Relationship to partial derivatives and Jacobian matrices. - https://youtu.be/_zp9exSS4XI
Mean Value Inequality. Fréchet derivatives in infinite dimension: some examples. - https://youtu.be/hZYWdFZU2y8
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
Banach's Fixed Point Theorem and Application to ODEs - https://youtu.be/AZ__ZY5IG8Q
The Inverse Function Theorem, Part 1 - https://youtu.be/aQ9oxoGLw6s
The Inverse Function Theorem, Part 2 - https://youtu.be/Rs1smbGIHhs
The Implicit Function Theorem - https://youtu.be/c7Hdl-s0Rdg
Introduction to submanifolds of R^d: definitions and the tangent space - https://youtu.be/9_L58C9UWKs
Implicitly defined submanifolds and a bit of Optimization - https://youtu.be/8nH1oAXFdYU
Some interesting examples (TBA)
Important: instructions for HW assignments have been updated.
First HW assignment (week of March 13th) - Solutions
Second HW assignment (weeks of March 18th and 25th) - Solutions
Third HW assignment (week of April 1st)
Fourth HW assignment (week of April 8th -- CORRECTED on Apr 16th)
Fifth HW assignment (week of April 23rd).
Sixth HW assignment (week of April 30th).
Seventh HW assignment (week of May 20th).
Eighth HW assignment (week of May 27th).
Ninth HW assignment (week of June 12th).
Tenth HW assignment (week of June 17th).
Lista extra (additional HW assignment for practice only -- Apr 11th)
Exercises on point-set topology