Math 108a: Classical Analysis

Instructor: Anton Gorodetski
Office: 282 Sloan
Phone: 395-4350
E-mail: asgor@caltech.edu
Office Hours: Friday, 12-1 p.m., and by appointment.
 

Teaching Assistant: Dmitry Pavlov
Office:  158 Sloan
E-mail: dmitry@caltech.edu
Office Hours: Sunday, 7-8 p.m.

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.
• Connectedness. Cantor sets and their properties. 
• 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. Inverse Function Theorem.
• Fractals: examples, fractal dimensions.

Prerequisites: Calculus of one variable, Ma2.

Main texts:

• N.L.Carothers, Real Analysis (Required).
• W. Rudin, Principles of Mathematical Analysis. Third edition (Recommended).

Additional references will be given for a few topics not covered by these books.

Grading: Weekly homework 50%, midterm exam 10%, final exam 40%.

Collaboration Policy: You may discuss homework problems with other students, but solutions should be written up individually in your own words. Take-home exams must be your own work.

  Blaise Pascal (1623-1662):
        We are usually convinced more easily by reasons we have found ourselves than by those which have occurred to others.

Homework

• Homework #1                    Solutions (HW#1)
• Homework #2                    Solutions (HW#2)
• Homework #3                    Solutions (HW#3)
• Midterm                             Solutions (midterm)
• Results of  Midterm:   from 45 to 50  - 28%
                                   from 36 to 45  - 15%
                                   from 25 to 36  - 37%
                                   less than 25     - 20%
• Homework #4                    Solutions (HW#4)
• Homework #5                    Solutions (HW#5)
• Homework #6                    Solutions (HW#6)
• Homework #7                    Solutions (HW#7)
• Final                           Solutions
• Results of Final:         from 45 to 50  - 18%
                                   from 40 to 44  - 29%
                                   from 35 to 39  - 15%
                                   from 30 to 34  - 23%
                                   from 25 to 29  - 10%
                                   less than 25     - 5%
• COURSE   SURVEY