Math 361: Real and Abstract Analysis
Fall 2012, Professor Levandosky
Exam 2 Solutions
1. For each pair of metric spaces A and B (with the usual Euclidean metric), either nd a
homeomorphism between them or prove that no homeomorphism exists between them.
(
Math 361: Real and Abstract Analysis
Fall 2012, Professor Levandosky
Solutions to Final Exam Practice Questions
1. For each sequence of functions fn , determine the pointwise limit function f and whether ir not fn
converges uniformly to f on [0, 1].
(a) f
Math 361: Real and Abstract Analysis
Fall 2012, Professor Levandosky
Exam 2 Practice Problems: Solutions
1. Suppose f : M R is continuous and A is a dense subset of M (cl(A) = M ). Decide whether each
statement below is true or false. If true, provide a p
Math 361: Real and Abstract Analysis
Fall 2012, Professor Levandosky
Exam 1 Practice Questions: Solutions
1. Determine which of the following are metrics. For those that are metrics, prove it, and for those that
are not, demonstrate that one of the metric
Math 362: Real and Abstract Analysis
College of the Holy Cross, Spring 2005
Homework 7, Due Friday, April 29
1. (a) Show that if f is absolutely continuous, then f is uniformly continuous.
(b) Give an example of a uniformly continuous function that is not
Math 362: Real and Abstract Analysis
College of the Holy Cross, Spring 2005
Homework 6, Due Friday, April 1
1. Suppose f =
Show that
n=1
fn , where fn is a sequence of nonnegative measurable functions.
fn .
f=
n=1
2. Let fn be a sequence of L1 functions s
Math 362: Real and Abstract Analysis
College of the Holy Cross, Spring 2005
Homework 5, Due Friday, March 18
1. We proved in class that
f +g =
f+
g
and
cf = c
f
for nonnegative measurable functions f and g and nonnegative constants c. Using the
denition
f
Math 362: Real and Abstract Analysis
College of the Holy Cross, Spring 2005
Homework 4, Due Friday, March 4
1. Let A = Q [0, 1] and B = Qc [0, 1]. Show that A and B are measurable, and nd
m(A) and m(B ). Show that A is measurable and nd its measure.
2. Sh
Math 362: Real and Abstract Analysis
College of the Holy Cross, Spring 2005
Homework 3, Due Friday, February 11
1. Let Inv be the inverse operator Inv(X ) = X 1 , dened on the space of n n invertible
matrices. Show that Inv is dierentiable with derivative
Math 362: Real and Abstract Analysis
College of the Holy Cross, Spring 2005
Homework 2, Due Friday, February 4
1. Suppose U is a connected open subset of Rn and f : U Rm is twice dierentiable,
with (D2 f )p = 0 for all p U . What can you say about f ?
2.
Math 362: Real and Abstract Analysis
College of the Holy Cross, Spring 2005
Homework 1, Due Friday, January 28
1
1
1. Let T : R2 R3 be the linear transformation with matrix A = 1 1. Compute
1 1
2
T , and nd a vector in v R such that T (v ) = T |v |.
2. Fi
Math 362: Real and Abstract Analysis
College of the Holy Cross, Spring 2005
Exam 2, Due Wednesday, April 13
Professor Levandosky
You have one week to work on the test. You may not consult with anyone (except me)
about the exam. You may use your notes, the
Math 362: Real and Abstract Analysis
College of the Holy Cross, Spring 2005
Exam 1, Due Wednesday, February 23
Professor Levandosky
You have one week to work on the test. You may not consult with anyone (except me)
about the exam. You may use your notes a
Math 361: Real and Abstract Analysis
Fall 2012, Professor Levandosky
Exam 1 Solutions
1. Let M = R, d(x, y ) = |ex ey |.
(a) Show that d is a metric on M .
Solution. d(x, y ) = |ex ey | 0 for all x, y M . It is clear that d(x, y ) = 0 if
x = y . If d(x, y