MT322 Analysis
 Spring 2015

Dan Chambers   365 Carney

chambers@bc.edu    617-552-3769 (email is more reliable)


Office Hours: Chambers: M3-4, W12-1, F1-2; Mustafa: TH11-1, F10-11.

This is a followup to MT321, Analysis I. Building on that material, we will cover integral calculus (Riemann and Riemann-Stieltjes integrals), sequences of functions (including uniform convergence of these), the Stone-Weierstrass theorem, power series, exponential and logarithmic functions, Fourier series, linear transformations and derivatives of functions from R^m into R^n, and an introduction to Lebesgue measure and integration.

The text we'll continue to use is Principles of Mathematical Analysis, third edition, by Walter Rudin.


Here is a link to the syllabus.

This is a work in progress; check back for updates.



Date
topic/section
HW assigned
HW due/solutions
January



12
definition of Riemann and R-S integrals
a non-integrable function

 

14
examples of R-S integrals
upper and lower sum and refinement results
central tool of R-S integrability
HW#1

16
continuity implies R-S integrability

19
MLK Day


21
monotonicity and integrability
piecewise cts implies integrability
compositions


23
an example
properties of R-S integrals
HW#2
HW#1
26
products of integrable functions
R-S integral wrt a step function


28
snow day- BC closed


30
R-S integral wrt a differentiable function
start of FTC, I


February



2
snow day- BC closed

4
FTC, integration by parts HW#3 HW#2
6
Schwarz inequality, odds and ends

9
snow day- BC closed

11
sequences and series of functions intro

HW#3
13
uniform convergence, examples, tools


16
Cauchy criterion, etc.
HW#4

18
uniform convergence and integrability


20
application; uniform convergence and continuity (metric space setting)

23
R(X) is a complete metric space
uniform convergence and differentiability

HW#4
25
midterm 1  solutions


27
more on uniform convergence and differentiability
a continuous everywhere, differentiable nowhere function


March



2
spring break


4
spring break


6
spring break


9
uniform boundedness and equicontinuity of a
sequence of functions
HW#5

11
convergence of a subsequence of functions

13
approximating a function;
Bernstein polynomials


16
proof of Weierstrass's approximation theorem
HW#5
18
power series introduction


20
Taylor/MacLaurin series
HW#6

23
remainder term; a MacLaurin series that converges to its function only at 0


25
convergence of M. series to the function; exp and log functions


27
E(x), L(x)

HW#6
30
class canceled


April



1
class canceled


3
Easter break


6
Easter break


8
why does L(E(x))=x? a series answer
start of orthonormal sets, inner product spaces
HW#7

10
Fourier series intro


13
F. series examples, convergence

HW#7
15
midterm 2  solutions


17
F. series on other intervals; convergence proof


20
Patriots' Day

22
sigma algebras on a set


24
Lebesgue measurable sets


27
Lebesgue measurable functions


29
Lebesgue integration  sample problems solutions


May



6
final exam 12:30 pm