Math508, Fall 2010
Jerry L. Kazdan
Problem Set 1
D UE : Thurs. Sept. 16, 2010. Late papers will be accepted until 1:00 PM Friday.
1. Let x0 = 1 and dene xk :=
increasing.
2. Show that 1 +
3xk1 + 4, k = 1, 2, . . . . Show that xk < 4 and that the xk are
1
ADVANCED ANALYSIS
MATH 360-361 & 508-509
A. A. KIRILLOV
1. Introduction
The goal of these lectures is to give a short and self-contained exposition
of basic facts of Analysis with accurate definitions and complete proofs.
Many people say that they hate ma
Math 508
December 9, 2010
Exam 2
Jerry L. Kazdan
9:00 10:20
Directions This exam has three parts, Part A asks for 3 examples (5 points each, so 15 points).
Part B has 4 shorter problems (8 points each so 32 points). Part C has 4 traditional problems (15
p
Exam 2
Math 508
December 9, 2010
Jerry L. Kazdan
9:00 10:20
Directions This exam has three parts, Part A asks for 3 examples (5 points each, so 15 points).
Part B has 4 shorter problems (8 points each so 32 points). Part C has 4 traditional problems (15
p
Math 508 October 14, 2010
Exam 1
Jerry L. Kazdan 9:00 10:20
Directions This exam has three parts, Part A asks for 4 examples (20 points, 5 points each). Part B has 4 shorter problems (36 points, 9 points each. Part C has 3 traditional problems (45 points,
1
Exam 2
Math 508
December 4, 2008
Jerry L. Kazdan
10:30 11:50
Directions This exam has two parts, Part A has 10 True-False problems (30 points, 3 points
each). Part B has 5 traditional problems (70 points, 14 points each).
Closed book, no calculators or
Math 508 December 4, 2008
Exam 2
Jerry L. Kazdan 10:30 11:50
Directions This exam has two parts, Part A has 10 True-False problems (30 points, 3 points each). Part B has 5 traditional problems (70 points, 14 points each). Closed book, no calculators or co
Exam 1
Math 508
October 16, 2008
Jerry L. Kazdan
10:30 11:50
Directions This exam has three parts, Part A has 4 problems asking for Examples (20 points, 5
points each), Part B asks you to describe some sets (20 points), Part C has 4 traditional problems
(
Math 508 December 8, 2006
Exam 2
Jerry L. Kazdan 12:00 1:20
Directions This exam has two parts, Part A has 3 shorter problems (8 points each, so 24 points), Part B has 5 traditional problems (15 points each, so 75 points). Closed book, no calculators but
Signature Math 508 December 8, 2006
Printed Name
Exam 2
Jerry L. Kazdan 12:00 1:20
Directions This exam has two parts, Part A has 3 shorter problems (8 points each, so 24 points), Part B has 5 traditional problems (15 points each, so 75 points). Closed bo
Exam 1
Math 508
October 12, 2006
Jerry L. Kazdan
12:00 1:20
Directions This exam has three parts, Part A has 4 problems asking for Examples (20 points, 5
points each), Part B asks you to describe some sets (20 points), Part C has 4 traditional problems
(6
Exam 1
Math 508
October 12, 2006
Jerry L. Kazdan
12:00 1:20
Directions This exam has three parts, Part A has 4 problems asking for Examples (20 points, 5
points each), Part B asks you to describe some sets (20 points), Part C has 4 traditional problems
(6
Math508, Fall 2010
Jerry L. Kazdan
Problem Set 8
D UE : Thurs. Nov. 11, 2010. Late papers will be accepted until 1:00 PM Friday.
Note: We say a function is smooth if its derivatives of all orders exist and are continuous.
1. a) Let A(t ) and B(t ) be n n
Math508, Fall 2010
Jerry L. Kazdan
Problem Set 7
D UE : Thurs. Nov. 4, 2010. Late papers will be accepted until 1:00 PM Friday.
Note: We say a function is smooth if its derivatives of all orders exist and are continuous.
1. Let f : [a, ) R be a smooth fun
Math508, Fall 2010
Jerry L. Kazdan
Problem Set 6
D UE : Thurs. Oct. 28, 2010. Late papers will be accepted until 1:00 PM Friday.
1. Give examples of the following:
a) An open cover of cfw_x R : 0 < x 1 that has no nite sub-cover.
b) A metric space having
Math508, Fall 2010
Jerry L. Kazdan
Problem Set 0: Rust Remover
D UE : These problems will not be collected.
You should already have the techniques to do these problems, although they may take some
thinking.
1. Show that for any positive integer n , the nu
Math508, Fall 2010
Jerry L. Kazdan
Problem Set 1
D UE : Thurs. Sept. 16, 2010. Late papers will be accepted until 1:00 PM Friday.
1. Let x0 = 1 and dene xk :=
increasing.
2. Show that 1 +
3xk1 + 4, k = 1, 2, . . . . Show that xk < 4 and that the xk are
1
MATH508. ADVANCED CALCULUS
LECTURE 9.ELEMENTS OF FUNCTIONAL ANALYSIS
A. A. KIRILLOV
1. The space of continuous functions
1.1. Completness. Let X be a metric (or a topological) space. Consider
the set C(X) of all continuous real-valued functions on X. It h
LECTURE 8. ELEMENTARY FUNCTIONS.
A. A. KIRILLOV
Here we give the rigorous definition and prove the basic properties of
so-called elementary functions.
1. Introduction
Traditionally the list of elementary functions include exponential, logarithmic, power,
HANDOUT 1
A. A. KIRILLOV
This handout contains some additional information for lectures 5 (Sequences) and 6 (Elements of topology).
1. Series
A series is an expression of the form
(1)
a1 + a2 + + an + . . .
where an are numbers, or, more generally, elemen
Math 508. Fall 2016
A.A.Kirillov
Dec 2016
Practice exam. Due Dec 1. Total 65 pts (Not included in grading)
10 points for each of the 5 non starred problems,
5 bonus points for each of the 3 starred problems.
1. Give the definition of the real number ( 2
Math508, Fall 2010
Jerry L. Kazdan
Problem Set 11
D UE : Never
Note: We say a function is smooth if its derivatives of ball orders exist and are continuous.
1. Partition [a, b] R into sub-intervals a < x1 < x2 < < xn = b . A function h(x) that is
constant
Math508, Fall 2010
Jerry L. Kazdan
Problem Set 10
D UE : Tues. Nov. 30, 2010. Late papers will be accepted until 1:00 PM Wednesday.
Note: We say a function is smooth if its derivatives of ball orders exist and are continuous.
1. Find an integer N so thst
Math508, Fall 2010
Jerry L. Kazdan
Problem Set 9
D UE : Thurs. Nov. 18, 2010. Late papers will be accepted until 1:00 PM Friday.
Note: We say a function is smooth if its derivatives of all orders exist and are continuous.
1. Let f (x) be a smooth function
Math508, Fall 2010
Jerry L. Kazdan
Problem Set 8
D UE : Thurs. Nov. 11, 2010. Late papers will be accepted until 1:00 PM Friday.
Note: We say a function is smooth if its derivatives of all orders exist and are continuous.
1. a) Let A(t ) and B(t ) be n n
Math508, Fall 2010
Jerry L. Kazdan
Problem Set 7
D UE : Thurs. Nov. 4, 2010. Late papers will be accepted until 1:00 PM Friday.
Note: We say a function is smooth if its derivatives of all orders exist and are continuous.
1. Let f : [a, ) R be a smooth fun
Math508, Fall 2010
Jerry L. Kazdan
Problem Set 6
D UE : Thurs. Oct. 28, 2010. Late papers will be accepted until 1:00 PM Friday.
1. Give examples of the following:
a) An open cover of cfw_x R : 0 < x 1 that has no nite sub-cover.
b) A metric space having
Math508, Fall 2010
Jerry L. Kazdan
Problem Set 5
D UE : Thurs. Oct. 21, 2010. Late papers will be accepted until 1:00 PM Friday.
k
1. [Ratio Test] Let ak be a sequence of complex numbers. Ley s := lim sup aa+1 . By comparison
k
with a geometric series, sh
Math508, Fall 2010
Jerry L. Kazdan
Problem Set 4
D UE : Thurs. Oct 7, 2010. Late papers will be accepted until 1:00 PM Friday.
5n + 17
.
n+2
3n2 2n + 17
. Calculate lim an .
b) Let an := 2
n
n + 21n + 2
1. a) Calculate lim
n
2. Investigate the convergence
Math508, Fall 2010
Jerry L. Kazdan
Problem Set 3
D UE : Thurs. Sept. 30, 2010. Late papers will be accepted until 1:00 PM Friday.
1. Find all (complex) roots z = x + iy of z2 = i .
2. Let xn > 0 be a sequence of real numbers with the property that they co