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University of
Final Examination
WATERLOO
Fall 2011
Name (Please Print):
UW Student Identication Number:
AMath 331 / PMath 331
Applied Real Analysis
Instructor: Kenneth R. Davidson
Date: Tuesday, December 13, 2011
Section: 001
Time: 7:30 p.m. to 10:00 p.m.
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Friday, November 9 Lecture 26 : Finite dimensional normed vector spaces. (Refers
to section 7.3 of your text)
1. Give a continuous map from n to a finite dimensional normed vector space V
which is 11 and onto.
Expectations:
26.1 Recall Recall that a fini
PMath/AMath 331, Fall 2013  Solutions for assignment 5
Handed out on Friday October 11; due on Friday October 18.
Topics: Open and closed subsets of a normed vector space.
Problem 1. Show by using the denition of closed set that the set S = cfw_x Rn :
1
PMath/AMath 331, Fall 2013 Solutions for assignment 4
Handed out on Friday October 4; due on Friday, October 11.
Topics: Normed vector spaces, convergence and completeness in Rn .
Problem 1 a) Show that if that lim xn = a then lim xn = a .
n
n
Solution.
S
PMath/AMath 331, Fall 2013  Assignment 6
Handed out on Friday October 18; due on Friday October 25.
Topics: Compact sets, HeineBorel theorem, continuous functions on Rn .
Submit questions 1 to 4, 6, 7. Bonus problems for extra points.
Problem 1. Determi
PMath/AMath 331, Fall 2013  Solutions for assignment 3
Handed out on Friday, September 27; due on Friday, October 4.
Topics: Cauchy sequences, Contractive sequences, Euclidean norm on Rn .
Problem 1. If x1 > 0 and xn+1 =
contractive and calculate its lim
PMath/AMath 331, Fall 2013  Assignment 8
Handed out on Friday November 1; due on Friday November 8.
Topics: Contractions, properties of continuous functions on Rn , the Extreme value
theorem.
Problem 1. Consider the linear transformation on R4 given by t
PMath/AMath 331, Fall 2013  Assignment 7
Handed out on Friday October 25. Due Friday November 1.
Topics: Lipschitz functions, linear transformations, contractions.
Only submit problems 1, 2, 4, 7 and (if you feel up to it) the Bonus problem.
Problem 3, 5
PMath/AMath 331, Fall 2013  Solutions for assignment 7
Handed out on Friday October 25. Due Friday November 1.
Topics: Lipschitz functions, linear transformations, contractions.
Only submit problems 1, 2, 4, 7 and (if you feel up to it) the Bonus problem
PMath/AMath 331, Fall 2013  Assignment 8
Handed out on Friday November 1; due on Friday November 8.
Topics: Contractions, properties of continuous functions on Rn , the Extreme value
theorem.
Problem 1. Consider the linear transformation on R4 given by t
PMath/AMath 331, Fall 2013  Assignment 6
Handed out on Friday October 18; due on Friday October 25.
Topics: Compact sets, HeineBorel theorem, continuous functions on Rn .
Problem 1. Determine whether the following subsets are compact or not. Justify you
PMath/AMath 331, Fall 2013  Assignment 9
Handed out on Friday November 8; due on Friday November 15.
Topics: Intermediate Value theorem, nite dimensional normed vector spaces, uniformly
continuous functions.
Problem 1. (a) Show that the function f (x) =
PMath/AMath 331, Fall 2013  Solutions for assignment 10
Handed out on Friday November 15; due on Friday November 22.
Topics: Pointwise convergence of functions, uniform convergence of functions.
Problem 1. Consider the sequence cfw_fn of functions on R
PMath/AMath 331, Fall 2013  Assignment 4.
Handed out on Friday October 4; due on Friday, October 11.
Topics: Normed vector spaces, convergence and completeness in Rn .
Problem 1
a) Show that if that lim xn = a then lim xn = a .
n
n
b) Show by example tha
PMath/AMath 331, Fall 2013  Assignment 5
Handed out on Friday October 11; due on Friday October 18.
Topics: Open and closed subsets of a normed vector space.
Problem 1. Show by using the denition of closed set that the set S = cfw_x Rn :
1 is closed in R
Friday, November 9 Lecture 26 : Finite dimensional normed vector spaces. (Refers
to section 7.3 of your text)
1. Give a continuous map from n to a finite dimensional normed vector space V
which is 11 and onto.
Expectations:
26.1 Recall Recall that a fini
Monday, November 12 Lecture 27 : The generalized HeineBorel theorem. (Refers
to section 7.3 of your text)
Expectations:
1. State the Generalized HeineBorel theorem for finite dimensional normed vector
spaces.
2. Recognize that, if W is a finite dimensio
Wednesday, November 14 Lecture 28 : Limits of sequences of functions. (Refers to
section 8.1)
1. Define pointwise convergence of a sequence of functions in C(n, m) .
2. Define uniform convergence of a sequence of functions in C(n, m).
3. Find the uniform
Friday, November 16 Lecture 29 : Some consequences of uniform convergence.
(Refers to 8.2 and 8.3)
Expectations:
1. Recognize that the limit of a uniformly converging sequence of continuous
functions must be continuous.
2. Recognize that the set of all re
AM/PM 331 Midterm information
When: Midterm is on Thursday November 1, 7pm to 9pm \
Location: MC 2054 & MC 2038

Alphabetical block A P : MC 2054
Alphabetical block Q Z : MC 2038
What: The midterm will cover material presented from lecture 1 to 17 inclus
Quick reference list of more important definitions and theorem statements.
1.1 Definition A definition of the Real numbers . In what follows, + denotes the positive integers cfw_1, 2, 3, .
The real numbers are defined as the set of all infinite decimal ex
PMath/AMath 331, Fall 2012  Solutions for assignment 10
Handed out on Friday November 16; due on Friday November 23.
Topics: Pointwise convergence of functions, uniform convergence of functions.
Problem 1. Consider the sequence cfw_fn of functions on R
PMath/AMath 331, Fall 2012  Solutions for assignment 9
Handed out on Friday November 9; due on Friday November 16.
Topics: Intermediate Value theorem, topology in normed vector spaces, uniformly
continuous functions, continuous functions on normed vector
PMath/AMath 331, Fall 2012  Assignment 8
Handed out on Friday November 2; due on Friday November 9.
Topics: Contractions, properties of continuous functions on Rn , the Extreme value
theorem.
Problem 1. Consider the linear transformation on R4 given by t
PMath/AMath 331, Fall 2013  Assignment 3
Handed out on Friday, September 27; due on Friday, October 4.
Topics: Cauchy sequences, Contractive sequences, Euclidean norm on Rn .
Problem 1. If x1 > 0 and xn+1 =
contractive and calculate its limit.
1
2 + xn
f
PMath/AMath 331, Fall 2013  Assignment 1
Handed out on Friday, September 13. Selected questions are due on Friday, September 20
Topics: Denition of limit, Properties of limits, Bounded sequences, Squeeze theorem.
You may invoke statements given in the le
PMath/AMath 331, Fall 2013  Solutions for assignment 2
Handed out on Friday, September 20; due on Friday, September 27.
Topics: Upper and lower bounds, Least upper bound principle, Monotone convergence,
Bolzano Weierstrass theorem, Cauchy sequences.
Subm