MT322 Analysis
 Spring 2014

Dan Chambers   365 Carney

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


Office Hours: Mondays 12-1, Wednesday 2-3, Fridays 11-12, or by appointment.

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. Time permitting, we will look at further topics, such as Fourier series or Lebesgue 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



13
definition of Riemann and R-S integrals
a non-integrable function

 

15
examples of R-S integrals
upper and lower sum and refinement results
central tool of R-S integrability


17
continuity implies R-S integrability

20
MLK Day


22
monotonicity and integrability
piecewise cts implies integrability
compositions
HW#1

24
an example
properties of R-S integrals


27
products of integrable functions
R-S integral wrt a step function


29
R-S integral wrt a differentiable function
start of FTC, I

HW#1
31
FTC, integration by parts, Schwarz inequality
HW#2

February



3
sequences and series of functions, introduction


5
snow day- BC canceled


7
uniform convergence- sequences


10
uniform convergence- series; Abel's Theorem
HW#3
HW#2
12
uniform convergence and integrability
alternating harmonic series and \pi/4


14
uniform convergence and continuity
sup norm


17
R(X) as a complete metric space
uniform convergence and differentiability

HW#3
19
exam 1  solns


21
more on uniform convergence and differentiability

24
a continuous everywhere, differentiable nowhere function


26
uniform boundedness and equicontinuity of a
sequence of functions


28
convergence of a subsequence of functions HW#4

March



3
spring break


5
spring break


7
spring break


10
approximating a function;
Bernstein polynomials


12
proof of Weierstrass's approximation theorem


14
conclusion of proof; power series introduction

HW#4
17
power series: continuity, differentiability, integrability results
a series representation for pi


19
more power series manipulations
intro to Taylor/MacLaurin series
HW#5

21
functions and their T/M series


24
convergence of the M. series of f to f


26
E(x) vs e^x, L(x) vs ln(x)
HW#6
HW#5
28
why does L(E(x))=x? a series answer
start of orthonormal sets, inner product spaces


31
inner product spaces, C-S inequality


April



2
start of Fourier series

HW#6
4
exam 2  solns


7
Fourier series, examples, convergence


9
F. series wrap up;
functions of several variables: linear transformations


11
linear transformations and differentiation

14
differentiation continued


16
inverse function theorem


18
Easter break

21
Patriots' Day HW#7

23
Lebesgue theory, sigma-algebras, measurable sets


25
measurable sets, measurable functions


28
measurable functions

HW#7
30
Lebesgue integrals

May



7
final exam 12:30 pm