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
Math 508 Fall 2014
Jerry Kazdan
Compactness
In these notes we will assume all sets are in a metric space X. These proofs are merely a
rephrasing of this in Rudin but perhaps the differences in wording
Math508, Fall 2010
Jerry L. Kazdan
Problem Set 2
D UE : Thurs. Sept. 23, 2010. Late papers will be accepted until 1:00 PM Friday.
1. Let F be a eld, such as the reals or the integers mod 7 and x, y F
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
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
thi
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
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
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 poi
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 p
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 poi
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 som
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 point
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 traditi
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
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
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 exis
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
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 +
Math 508, Fall 2014
Jerry L. Kazdan
The Archimedean Property
Definition An ordered field F has the Archimedean Property if, given any positive x and
y in F there is an integer n > 0 so that nx > y.
Th
Basic Examples
A = cfw_x R : x = 1, 2, 3, 4
B1 = cfw_x R : 0 < x < 1
B2 = cfw_(x, y) R2 : 0 < x < 1, y = 0
C = cfw_x R : 0 x 1
D = cfw_x R : x = 1, 2, 3, 4, . . .
E = cfw_x R : x = 1, 1/2, 1/3, . . .
Math 508, Fall 2014
Jerry L. Kazdan
Aug. 28, 2014
TEXand LATEX: Learn this.
Problem Set 0
get stuck
work together
definitions
proofs: The best proof one you thought of yourself.
algebra, geometry
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
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
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 element
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 +
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 pr
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
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
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 exis
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 exis