Date 
topic/section 
HW assigned 
HW due/solutions 
January 

13 
definition of Riemann and RS
integrals a nonintegrable function 


15 
examples of RS integrals upper and lower sum and refinement results central tool of RS integrability 

17 
continuity implies RS integrability  
20 
MLK Day 

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

24 
an example properties of RS integrals 

27 
products of integrable functions RS integral wrt a step function 

29 
RS 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, CS
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, sigmaalgebras,
measurable sets 

25 
measurable sets, measurable
functions 

28 
measurable functions 
HW#7 

30 
Lebesgue integrals  
May 

7 
final exam 12:30 pm 