Real Variables II  Math 321
Spring 2015
Instructor: Malabika Pramanik
Mathematics Building, Room 214
Phone: (604)8222855
Email: [email protected]
Oce hours: To be announced on the course website.
Web page : The course website is
http:/www.math.
Homework 7  Math 321, Spring 2015
Due on Friday March 13
1. Recall Jordans theorem: a function f : [a, b] R is of bounded variation if and only if f
can be written as the dierence of two nondecreasing functions g and h.
(a) Show that the decomposition f
Homework 6 Solutions  Math 321, Spring 2015
1. Given a nonconstant nondecreasing function : [a, b] R, let R [a, b] denote the collection
of all bounded functions on [a, b] which are RiemannStieltjes integrable with respect to .
Is R [a, b] a vector spa
Homework 6  Math 321, Spring 2015
Due on Friday February 27
1. Given a nonconstant nondecreasing function : [a, b] R, let R [a, b] denote the collection
of all bounded functions on [a, b] which are RiemannStieltjes integrable with respect to .
Is R [a,
Homework 5 Solutions Math 321, Spring 2015
1. The classical Weierstrass approximation theorem says that the class of polynomials is dense
in C[a, b]. The StoneWeierstrass theorem, on the other hand, provides a necessary and
sucient condition for a subal
Homework 3 Solutions  Math 321,Spring 2015
(1) 1a. We want to nd a sequence cfw_pn of polynomial such that pn f uniformly
on any bounded subset of R. By The First Weierstrass Approximation Theorem,for
each positive integer N , we can nd a sequence of po
Homework 4 Solutions  Math 321, Spring 2015
1. Weierstrasss second theorem states that any continuous 2periodic function f on R is
uniformly approximable by trigonometric polynomials. The aim of this exercise is to prove
this statement.
(a) Deduce Weier
Homework 5  Math 321, Spring 2015
Due on Friday February 13
1. The classical Weierstrass approximation theorem says that the class of polynomials is dense
in C[a, b]. The StoneWeierstrass theorem, on the other hand, provides a necessary and
sucient cond
Homework 4  Math 321, Spring 2015
Due on Friday February 6
1. Weierstrasss second theorem states that any continuous 2periodic function f on R is
uniformly approximable by trigonometric polynomials. The aim of this exercise is to prove
this statement.
(
Homework 7 Solutions  Math 321, Spring 2015
1. Recall Jordans theorem: a function f : [a, b] R is of bounded variation if and only if f
can be written as the dierence of two nondecreasing functions g and h.
(a) Show that the decomposition f = g h is by
Homework 8  Math 321, Spring 2015
Due on Friday March 27
1. Hellys rst and second theorems were critical components of our proof of the Riesz representation theorem for continuous linear functionals on C[a, b]. You have already proved Hellys
rst theorem
Homework 8 Solutions
1. Hellys rst and second theorems were critical components of our proof of the Riesz representation theorem for continuous linear functionals on C[a, b]. You have already proved Hellys
rst theorem in a previous assignment. Prove the s
Review worksheet  Countability, density, separability
Math 321, Spring 2015
Two sets A and B are said to be equivalent if there exists a bijection f : A B, i.e.,
if f is a onetoone function from A onto B.
A set A is called nite if either A = or if A
Practice Problem Set 2  RiemannStieltjes integration
More problems may be added to this set. Stay tuned.
1. If R [a, b] contains all step functions on [a, b], show that is continuous.
2. Given a sequence cfw_xn of distinct points in (a, b) and a sequen
Practice Problem Set 1  Sequences and Series of functions
More problems may be added to this set. Stay tuned.
1. Evaluate with justication
lim
n
0
n + sin nx
dx.
3n sin2 nx
2. For each n N, you are given a dierentiable function fn : R R that satises
fn
Practice Problem Set for the nal exam
1. Let S denote the set of functions in C[, ] of the form
f (x) = a sin x + b sin 2x
where a and b are arbitrary real numbers. Let g(x) = x for x [, ]. Find f S for
which g f 2 is smallest.
(Answer: f (x) = 2 sin x
Midterm Solutions  Math 321
1. Give complete denitions of the following terms:
(a) an equicontinuous family of functions in C[0, 1].
Solution. A family of functions F C[0, 1] is said to be equicontinuous if for every
> 0, there exists > 0 such that
(1)

Midterm Review  Math 321, Spring 2015
1. Give an example of an equicontinuous family of nonconstant functions that is not totally
bounded.
Sketch of solution. The function class LipK [0, 1] \ cfw_constant functions for any xed K provides an example. Thi
Homework 9  Math 321, Spring 2015
1. Let f be a dierentiable 2periodic function with continuous rst derivative. Is f the uniform limit of its partial Fourier sums?
Solution. Yes. Using integration by parts and periodicity of f ,
1
f (k) =
2
f (t)e
ikt
1
Homework 9  Math 321, Spring 2015
Due on Wednesday April 8
1. Let f be a dierentiable 2periodic function with continuous rst derivative. Is f the
uniform limit of its partial Fourier sums?
2. Determine whether the following statement is true or false: I
Homework 3  Math 321, Spring 2015
Due on Friday January 30
1. (a) Show that if f is continuous on R, then there exists a sequence cfw_pn of polynomials
such that pn f uniformly on each bounded subset of R.
(b) Show that there does not exist a sequence o
Homework 1 Solutions  Math 321,Spring 2015
1a Pointwise convergence: for a x x [0, ) If x = 0, fn (x) = 0 for all n so it converges
to 0. If x = 0, we have
nx
= lim
n 1 + n2 x2
n
lim
x
n
1
n2
+
x2
=
0
=0
0 + x2
The convergence is not uniform on [0, ): If
Homework 2 Solutions  Math 321,Spring 2015
(1) For each a [0, 1] consider fa B[0, 1] i.e. fa : [0, 1] [0, 1] dened by
fa (t) =
1 if t = a
0 if t = a
There are uncountably many such fa as [0, 1] is uncountable. Also if a = b then
fa fb = 1 (if it is clear
MATH 321  HOMEWORK #4
Due Friday, Feb 26.
P ROBLEM 1. Let : [a, b] R2 be a continuous rectifiable curve. Prove that
L(x) = xa
is continuous on [a, b]. (Here xa is the length of on the interval [a, x].)
P ROBLEM 2. Find sequences of functions cfw_fn and
MATH 321  HOMEWORK #10
P ROBLEM 1. Consider the function f(x) on [, ]:
0 if x [, 0]
f(x) =
x if x (0, ]
(a) Find the Fourier series
f(x)
X
cn einx .
n
(b) Express the Fourier series in terms of sines and cosines.
(c) Evaluate the Fourier series at x = 0
MATH 321  PRACTICE EXAM #1
The first five problems are similar in length and difficulty to the problems in the actual
exam. Problem 6 is more difficult.
P ROBLEM 1. In each case below determine if the sequence or sum converges pointwise/
uniformly/neithe
MATH 321  PRACTICE EXAM #2
All functions take values in R.
P ROBLEM 1. Let cfw_fn be the sequence of continuous functions on [0, 1], such that
n
fn (x) = x sin , for x (0, 1].
x
Is the sequence equicontinuous? Prove it or explain why not.
P ROBLEM 2. Le
MATH 321: Real Variables II Notes
2015W2 Term
Taught by Dr. Kalle Karu, taken by Adrian She
Please report typos or errors to Adrian at [email protected]
Contents
I
RiemannSteiljes Integration
5
1 The Riemann Integral
1.1 Darbouxs Definition of the
Math 321 Final Exam
8:30am, Tuesday, April 20, 2010
Duration: 150 minutes
Name:
Student Number:
Do not open this test until instructed to do so! This exam should have 17 pages,
including this cover sheet. No textbooks, calculators, or other aids are allow
Math 321 Final Exam
Apr 20, 2009
Duration: 150 minutes
Student Number:
Name:
Section:
Do not open this test until instructed to do so! This exam should have 19 pages,
including this cover sheet. No textbooks, calculators, or other aids are allowed. Turn o