Análise no R^n

IMPA - Mar 3rd to Jul 3rd,2020

News

  • [20/03] First video is available below.
  • [17/03] Due to the COVID-19 pandemic, we are moving our class online. Instructions will be sent by email. Check the section on topics covered + handouts for weekly info. I will also post videos and weekly homework assignments.
  • [12/03] First homework assignment at bottom of page.
  • [10/03] Check this webpage often for news about the class.

General information

  • Instructor: Roberto Imbuzeiro Oliveira
  • TA (monitora): Cynthia Bortolotto
  • Class times: Wed and Thu from 10h30 to 12h in room 236, IMPA
  • Recitations/TAing sessions (aulas de monitoria): Tuesdays from 10h30 to 12h in room 236, IMPA

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.

  • Tests on April 8th 15th and June 3rd from 10h to 12h. All problems in the tests will come from the homework assignments.
  • Exams on May 6th and July 2nd from 9h to 12h. About half the problems in exams will come from the homework assignments.
  • Final numerical grade = (2/3)*MP + (1/3)*MAX(MT,MP), MT= average test grade and MP =average exam grade.
  • 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

We will cover most of my lecture notes.

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 + Handouts (NEW)

(Week of March 11th). 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.

(Week of March 18th). C([0,1],R) is complete with the sup norm. Equivalent metrics and norms. Definitions and many examples of continuous functions, including functions over R^n; functions defined in terms of distances; and (bounded) linear and multilinear mappings.

Videos (NEW)

[20/03/2020] C([0,1],R) is complete with sup norm - https://youtu.be/WpTcXkbw7Ks

[21/03/2020] Equivalent metrics and norms - https://youtu.be/w8L4JQurGpc

[23/03/2020] Continuity: basics - https://youtu.be/Tu2MO0mUa38

[24/03/2020] Lipschitz functions and functions related to distances - https://youtu.be/DK1X7LCQHf4

Homework assignments

Important: you are not to turn these in! Rather, these serve to prepare you for the tests and exams. (See above)

First HW assignment.

Second HW assignment (18/03/2020).