Math 5616, Spring '00

Last changed: May 10,'00

Math 5616: Honors: Introduction to Analysis II. Spring '00

The class meets MWF at 12:20 in VinH 301.
Office Hours 11:15 -- 12:05 MWF; "On call" after class; 9:05 -- 9:55 MWF (priority then for Wavelets class, Math 5467)
The Final Exam will be on Friday May 12, 4 - 6 pm, in VinH 301.

Announcements
I'll be in the office until noon today, Wednesday May 10.
I apologize for a delay in getting gradelines up: incompatible spreadsheet! My application at home is new, and the old one here won't read it! I have sent a message to the computer gurus in hopes they can quickly install the new version this morning.

See below for Final Exam Review Questions

Questions that have come up, relevant to the Final:
Riemann-sum approximations of the integral of x^n from 0 to x will only be asked for specific small values of n.

Why does AB = I (in the finite-dimensional case) imply BA = I?
First, by 9.5 in Rudin, a square matrix M defines a one-to-one linear transformation iff M defines a linear transformation that is onto.
Thus, if AB = I, both of A and B are one-to-one and onto.
Then AB - BA = I - BA, so A(AB - BA) = A(I - BA) = A - (AB)A = 0.
But A is one-to-one, so AB - BA must be zero. Otherwise, there would be some x \ne 0 such that y := (AB - BA)x \ne 0. But then 0 \ne Ay = A(AB - BA)x = 0, a contradiction.

Challenge problems

Challenge problem U Can 7.23 be changed so that we construct a uniformly convergent subsequence?

Challenge problem P For each positive even integer find an even polynomial P(x) of that degree that is 0 at 0 and such that the uniform norm of P(x)-x on [0, 1] is minimal, and show that these minimal norms tend to zero as the degrees tend to infinity.

Gradelines for Test 1: pseudotop=91; A=73; B=57; C=45; D=35.

The links here are to PDF documents. They require Adobe's Acrobat Reader.

Syllabus and Assignments

Math 5616 assignments.

Syllabus.

Supplemental Notes

Integer, rational, real powers of positive numbers: 04/11/00

Exponential and logarithmic functions.

Complex exponential and trig functions.

Intro: Riemann integral via Riemann sums

Two more criteria for Riemann integrability

Riemann-Stieltjes integrals: bounded variation