Real Variables II - Math 321
Spring 2015
Instructor: Malabika Pramanik
Mathematics Building, Room 214
Phone: (604)822-2855
Email: [email protected]
Oce hours: To be announced on the course web
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,f
Homework 4 Solutions - Math 321, Spring 2015
1. Weierstrasss second theorem states that any continuous 2-periodic function f on R is
uniformly approximable by trigonometric polynomials. The aim of thi
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 Stone-Weierstrass theorem, o
Homework 4 - Math 321, Spring 2015
Due on Friday February 6
1. Weierstrasss second theorem states that any continuous 2-periodic function f on R is
uniformly approximable by trigonometric polynomials.
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 subse
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
=
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 un
Homework 2 - Math 321, Spring 2015
Due on Friday January 23
1. Let B[0, 1] denote the space of all real-valued bounded functions on [0, 1], equipped with
the metric topology generated by the sup norm.
Homework 1 - Math 321, Spring 2015
Due on Friday January 16
1. (a) Does the sequence of functions
nx
(1 + n2 x2 )
converge pointwise on [0, )? Is the convergence uniform on this interval? If yes, give
Integration on Manifolds
Manifolds
A manifold is a generalization of a surface. We shall give the precise definition shortly.
These notes are intended to provide a lightning fast introduction to integ
Problem Solutions for Integration on Manifolds
Problem M.1 Let A be an atlas for the metric space M. Prove that there is a unique
maximal atlas for M that contains A.
Solution. Define A to be the set
The Contraction Mapping Theorem and
the Implicit Function Theorem
Theorem (The Contraction Mapping Theorem) Let Ba = ~x IRd k~xk < a
denote the open ball of radius a centred on the origin in IRd . I
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 Stone-Weierstrass theorem, on the other hand,
Homework 6 - Math 321, Spring 2015
Due on Friday February 27
1. Given a nonconstant non-decreasing function : [a, b] R, let R [a, b] denote the collection
of all bounded functions on [a, b] which are
Homework 6 Solutions - Math 321, Spring 2015
1. Given a nonconstant non-decreasing function : [a, b] R, let R [a, b] denote the collection
of all bounded functions on [a, b] which are Riemann-Stieltje
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 one-to-one function from A
Practice Problem Set 2 - Riemann-Stieltjes 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 seq
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 g
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 f
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 equicontin
Midterm Review - Math 321, Spring 2015
1. Give an example of an equicontinuous family of non-constant functions that is not totally
bounded.
Sketch of solution. The function class LipK [0, 1] \ cfw_co
Homework 9 - Math 321, Spring 2015
1. Let f be a dierentiable 2-periodic function with continuous rst derivative. Is f the uniform limit of its partial Fourier sums?
Solution. Yes. Using integration b
Homework 9 - Math 321, Spring 2015
Due on Wednesday April 8
1. Let f be a dierentiable 2-periodic function with continuous rst derivative. Is f the
uniform limit of its partial Fourier sums?
2. Determ
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 H
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
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 non-decreasing functions
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 non-decreasi
Euler Angles
Euler angles are three angles that provide coordinates on SO(3). Denote by R1 ()
and R3 () rotations about the x and z axes, respectively, by an angle . That is
1
0
R1 () = 0 cos
0 sin
SelfAdjoint Matrices Problem Solutions
Problem M.1 Let V Cn . Prove that V is a linear subspace of Cn .
Solution. If v, w V and C, then
hv + w, ui = hv, ui + hw, ui = 0 for all u V = v + w V
hv, ui =