An Introduction to Functional Analysis

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Course Date: 12 September 2014 to 07 November 2014 (8 weeks)

Price: free

Course Summary

Learn about functional analysis


Estimated Workload: 6-8 hours/week

Course Instructors

John Cagnol

John Cagnol is currently the Dean of Engineering and Associate Dean of Studies at Ecole Centrale Paris, an ivy-league engineering "grand-école".

He specializes in Partial Differential Equations (PDEs) and Modeling and Simulation and has authored over 30 research papers and edited over five proceedings volumes for various institutions.

John has taught and carried out research at the University of Virginia in the United States and has been the director of the mathematics department and laboratory of scientific computing at the ESILV in France.

As for his professional interests, John is involved with the IFIP (International Federation for Information Processing) and is the secretary of the working group on Control of Distributed Systems. He is also on the board of the ICAST (International Conference on Advancement of Science and Technology) and is an editor of "Smart Material and Structures", a publication of the IOP.

Anna Rozanova-Pierrat

Anna Rozanova-Pierrat is assistant professor in Applied Mathematics at École Centrale Paris. 
Her research areas include mathematical physics involving the models governed by partial different equations. She specializes in the theory behind inverse problems, equations of non-linear acoustics and diffusion problems on fractals, which are also subjects of her 10 scientific publications.

Course Description

Functional analysis is the branch of mathematics dealing with spaces of functions. It is a valuable tool in theoretical mathematics as well as engineering. It is at the very core of numerical simulation.

In this class, I will explain the concepts of convergence and talk about topology. You will understand the difference between strong convergence and weak convergence. You will also see how these two concepts can be used.

You will learn about different types of spaces including metric spaces, Banach Spaces, Hilbert Spaces and what property can be expected. You will see beautiful lemmas and theorems such as Riesz and Lax-Milgram and I will also describe Lp spaces, Sobolev spaces and provide a few details about PDEs, or Partial Differential Equations.

FAQ

  • Will I get a Statement of Accomplishment after completing this class?
    Yes. Students who successfully complete the class will receive a Statement of Accomplishment signed by the instructor.

  • What resources will I need for this class?
    For this course, all you need is an Internet connection and the time to view the videos, understand the material, discuss the material with fellow classmates, take the quizzes and solve the problems.

  • What pedagogy will be used?
    This MOOC is in English but the math will be taught with a "French Touch".

  • What does "teaching math with a French touch" mean?
    France has a long-standing tradition where math is addressed from a theoretical standpoint and studied for its implicit value throughout high school and preparatory school for the high-level entrance exams. This leads to a mindset based on proofs and abstraction. This mindset has consequences on problem solving that is sometimes referred to as the “French Engineer”. In contrast, other countries have a tradition where math is addressed as a computation tool.

  • Does it mean it will abstract and complicated?
    The approach will be rather abstract but I will be sure to emphasize the concepts over the technicalities. Above all, my aim is to help you understand the material and the beauty behind it.

Syllabus

Week 1: Topology; continuity and convergence of a sequence in a topological space.
Week 2: Metric and normed spaces; completeness
Week 3: Banach spaces; linear continuous functions; weak topology
Week 4: Hilbert spaces; The Riesz representation theorem
Week 5: The Lax-Milgram Lemma
Week 6: Lp spaces; Fischer-Riesz
Week 7: Sobolev spaces
Week 8: Use of functional analysis for Partial Differential Equations

Format

The class will consist of a series of lecture videos, usually between five and twelve minutes in length.  There will be approximately one hour worth of video content per week. Some of the videos contain integrated quiz questions. There will also be standalone quizzes that are not part of the video lectures; you will be asked to solve some problems and evaluate the solutions proposed by your fellow classmates. There will be a final exam.

There will be some additionnal contents in the form of PDF files.

Course Workload

6-8 hours/week

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