Math 108a
 
Classical Analysis
Fall 2014-15
 
MWF 11:00 AM // 151 Sloan 
Course Description | Policies | Textbooks | Lecture Notes | Topics Covered | Homework | Exams

Instructor:  Rupert Frank, 178 Sloan, rlfrank @ caltech.edu
Office Hours:  Wednesday, 3:00 - 4:00 pm


Lead TA:  Isak Mottleson, 379 Sloan, imottels @ caltech.edu
Office Hours:  Tuesday, 3:00 - 4:00 pm


TA:  Kevin Li, kkli @ caltech.edu


Feedback Form
Announcements
 

 




 

Course Description
  Top

Many of the results encountered in calculus seem to follow common themes, even though they are usually formulated as different topics. As an example think of the various notions of convergence typically discussed in a calculus class: limits of a sequence of numbers, point-wise versus uniform convergence of sequences of functions, Riemann integrability etc.

The interrelation does become very clear when viewing the objects under consideration as elements of more abstract spaces equipped with some ``notion of closeness'' (topology). Goal of this class is to develop some of the frameworks (``metric spaces'') allowing for this more general point of view, which will naturally lead to many of the questions of classical real analysis.

The first term of Ma 108 covers the following topics in real analysis:

  • Ordered sets, upper and lower bounds, least upper bound axiom, the construction of real numbers. 
  • Topology of metric spaces. Open, closed, bounded and compact sets,  perfect and connected sets. Sequences in metric spaces. 
  • The Heine-Borel Theorem. The Bolzano-Weierstrass Theorem. Compact metric spaces. 
  • Normed linear spaces. Completeness. Banach spaces. Contraction Mapping Principle and applications. 
  • Sequences and series of functions. Uniform convergence. Equicontinuity. The Arzela-Ascoli theorem. The Weierstrass approximation theorem. 
  • Bounded linear transformations. The derivative as a linear transformation. Taylor theorem in Rn. 


 

Prerequisites: Calculus of one and several variables.

 


Policies
  Top

Click here for details.
 

 




Textbooks
  Top


Textbook:
Principles of Mathematical Analysis, 3rd Edition, Walter Rudin, ISBN: 007054235X
 

Lecture Notes
  Top

Date Description

















Topics Covered
  Top

Date Description




















   
   
   
   

Homework
  Top
 
Due Date Homework  Solutions

Oct 9, 4:00 PM

Set no. 1


Oct 16, 4:00 PM

Set no. 2


Oct 23, 4:00 PM

Set no. 3


Oct 28, 4:00 PM

Set no. 4



 

Nov 13, 4:00 PM

Set no. 5


Nov 20, 4:00 PM

Set no. 6


Nov 26, 4:00 PM

Set no. 7, Part I


Dec 3, 4:00 PM

Set no. 7, Part II




Notes on implicit function theorem

































  | © California Institute of Technology | Questions?  kaubry @ caltech.edu