Math 8601: REAL ANALYSIS. Fall 2010 Problems for Final Exam on Saturday, December 18, 4pm6pm, VinH 1. This Final Exam will be based on the material from the textbook, in the following Sections: 0.50.6, 1.11.5 ,2.12.5, 2.6 (Theorems 2.40, 2.41), 3.13.2, 3.
Math 8601. December 18, 2010. Final Exam. Problems and Solutions. #1. Let A be an arbitrary set, and for each A, let an open ball B Rn be defined. Show that there is a finite or countable subset A0 A, such that B =
A A0
B .
Proof. The open set
:=
A
B =
j
Math 8601. October 6, 2010. Midterm Exam 1. Problems and Solutions. Problem 1. Let (X, M, ) be a measure space with (X) < . Show that for arbitrary A, B, C M, we have |(A B) - (A C)| (BC). where BC := (B \ C) (C \ B) = (B C) \ (B C) - the symmetric differ
Math 8601. November 17, 2010. Midterm Exam 2. Problems and Solutions. Problem 1. Find
n
lim
1 1 1 + + + . n n+1 2n - 1
Solution. After rewriting Sn := 1 1 1 1 + + + = n n+1 2n - 1 n
1 n-1
1 1+ k=0
k n
,
one can see that this is a Riemann sum for the integ
Math 8601/2: REAL ANALYSIS. Syllabus: FALL 2010
Class Times and Location: 10:10 am 11:00 am MWF, VinH 1. Instructor: Mikhail Safonov, VinH 231, tel: 625-8571, email: [email protected] http:/www.math.umn.edu/safonov Office Hours: MWF, 11:15 am 12:05 pm,
Math 8601
October 16, 2009
Name:
Midterm Exam
October 16, 2009
Closed book exam. Books, notes, and electronic devices may not be used.
(16) 1. Define each of the following statements or notation.
(4)
a. M is a -algebra on the set X .
(4)
b.
( X ,M )
(4)
c
Math 8601
Solutions to First Exam
Fall 2009
Midterm Exam Solutions
October 16, 2009
(16) 1. Define each of the following statements or notation.
(4)
a. M is a -algebra on the set X .
If M P ( X ) is nonempty and closed under complements and countable unio
Math 8601
November 23, 2009
Name:
Midterm Exam
November 23, 2009
Closed book exam. Books, notes, and electronic devices may not be used.
(20) 1.
(4)
a. Let and be signed measures on a measurable space ( X , M ) . Define .
(4)
b.
(4)
c. Let be a signed mea
Math 8601
Solutions to Second Exam
Fall 2009
Midterm Exam Solutions
November 23, 2009
(20) 1.
(4)
a. Let and be signed measures on a measurable space ( X , M ) . Define .
Solution. There exist E M and F M , with E F = and E F = X , such that E is
null for
Math 8601
December 18, 2009
Name:
Final Exam
December 18, 2009
Closed book exam. Books, notes, and electronic devices may not be used. Answers should be complete, concise,
and mathematically rigorous.
(40) 1. Define each of the following statements or not
Math 8602 Midterm Exam
April 10, 2009
Closed book exam. Books, notes, and electronic devices may not be used.
(24) 1. Define each of the following statements. You may assume that everyone knows the definition
of a linear space.
(4)
a.
X is a Hilbert space
Math 8602
May 14, 2009
Name:
Math 8602 Final Exam
May 14, 2009
Closed book exam. Books, notes, and electronic devices may not be used. Answers should be complete, concise,
and mathematically rigorous.
(40) 1. Define each of the following statements. You m
Math 8602
February 26, 2010
Name:
Math 8602 Midterm Exam
February 26, 2010
Closed book exam. Books, notes, and electronic devices may not be used.
(24) 1. Define each of the following statements. You may assume that everyone knows the definition of a
topo
Math 8602
Solutions to First Exam
Spring 2010
Midterm Exam Solutions
February 26, 2010
(24) 1. Define each of the following statements. You may assume that everyone knows the definition of a
topological space and a linear space.
(4)
a.
X is a locally comp
Math 8602 Midterm Exam
April 9, 2010
Closed book exam. Books, notes, and electronic devices may not be used.
(24) 1. Define each of the following statements. You may assume that everyone knows the definition
of a linear space.
(4)
a.
, is an inner produc
Math 8601: REAL ANALYSIS.
Fall 2010
Homework #7. Problems and Solutions. #1. Let f L1 (R1 ). Show that 1 Sn (x) = n in L1 (R1 ) as n , i.e. |Sn (x) - S(x)| dx 0 as n .
n-1
f x+
j=0
j S(x) = n
x+1
f (t) dt
x
Proof. By Theorem 2.26, > 0 there are a constant
Math 8601: REAL ANALYSIS.
Fall 2010
Homework #6. Problems and Solutions. #1 (Borel-Cantelli Lemma). Let (X, M, ) be a measure space, and cfw_An be a sequence of sets in M. Show that
if
n=1
(An ) < ,
then lim sup An = 0,
n
where lim sup An :=
n
An .
k=1
Math 8601: REAL ANALYSIS. Fall 2010 Some problems for Midterm Exam #1 on Wednesday, October 6. You will have 50 minutes (10:10 am11:00 am) to work on 5 problems, 2 of which will be selected from the following list. It is recommended to prepare solutions o
Math 8601: REAL ANALYSIS. Fall 2010 Problems for Midterm Exam #2 on Wednesday, November 17. This Midterm will be based on the material for the textbook up to (including) Section 2.3. You will have 50 minutes (10:10 am11:00 am) to work on 5 problems, 2 of
Math 8601: REAL ANALYSIS.
Fall 2010
Homework #1 (due on W, September 15). Updated on Sat, September 11. 50 points are divided between 5 problems, 10 points each. #1. Let F be a compact subset of Rn . Show that there are point x0 , y0 F , such that diamF :
Math 8601: REAL ANALYSIS.
Fall 2010
Homework #2 (due on W, September 29). 50 points are divided between 5 problems, 10 points each. #1. Let f (x) be a continuous function on [-1, 1], such that f (-1) < 0 < f (1). Show that f (c) = 0 for some c (-1, 1). Hi
Math 8601: REAL ANALYSIS.
Fall 2010
Homework #3 (due on W, October 13). 50 points are divided between 5 problems, 10 points each. #1. Let f be a real function on R1 . The image and the inverse image of a subset A R1 under f are correspondingly f (A) = cfw
Math 8601: REAL ANALYSIS.
Fall 2010
Homework #4 (due on Wednesday, October 27). 50 points are divided between 5 problems, 10 points each. #1. Let Rn be represented in the form Rn =
Ik , where cfw_Ik are non-overlapping cubes
k=1
with edge length 1. Let
Math 8601: REAL ANALYSIS.
Fall 2010
Homework #5 (due on Wednesday, November 10). 50 points are divided between 5 problems, 10 points each. #1 (Problem 30 on p.40. Let E be a Lebesgue measurable set in R1 with Lebesgue measure m(E) > 0. Show that for any <
Math 8601: REAL ANALYSIS.
Fall 2010
Homework #6 (due on Wednesday, November 24). 50 points are divided between 5 problems, 10 points each. #1 (Borel-Cantelli Lemma). Let (X, M, ) be a measure space, and cfw_An be a sequence of sets in M. Show that
if
n=1
Math 8601: REAL ANALYSIS.
Fall 2010
Homework #7 (due on Friday, December 10). 50 points are divided between 5 problems, 10 points each. #1. Let f L1 (R1 ). Show that 1 Sn (x) = n in L1 (R1 ) as n , i.e. |Sn (x) - S(x)| dx 0 as n . Hint. Using Theorem 2.26
Math 8601: REAL ANALYSIS.
Fall 2010
Homework #1. Problems and Solutions. #1. Let F be a compact subset of Rn . Show that there are point x0 , y0 F , such that diam F := supcfw_ |x - y| : x, y F = |x0 - y0 |. Proof. By definition of sup, there are sequenc
Math 8601: REAL ANALYSIS.
Fall 2010
Homework #2. Problems and Solutions. #1. Let f (x) be a continuous function on [-1, 1], such that f (-1) < 0 < f (1). Show that f (c) = 0 for some c (-1, 1). Proof. Take c := supcfw_x [-1, 1] : f (x) 0 (0, 1). We claim
Math 8601: REAL ANALYSIS.
Fall 2010
Homework #3. Problems and Solutions. #1. Let f be a real function on R1 . The image and the inverse image of a subset A R1 under f are correspondingly f (A) = cfw_y : y = f (x) for some x A, f -1 (A) = cfw_x : f (x) A.
Math 8601: REAL ANALYSIS.
Fall 2010
Homework #4. Problems and Solutions. #1. Let Rn be represented in the form Rn =
Ik , where cfw_Ik are non-overlapping cubes
k=1
with edge length 1. Let Fk be a closed subset of Ik , k = 1, 2, . . . Show that the set F