Math 245A Homework 1
Brett Hemenway October 19, 2005
3. a. Let M be an innite -algebra on a set X Let Bx be the intersection of all E M with x E . Suppose y Bx , then By Bx because Bx is a set in M which contains y . But Bx By because if y E for any E M,
Math 202A Homework 13
Roman Vaisberg November 28, 2007
Problem 17. Suppose f is dened on R2 as follows: f (x, y ) = an if n x < n + 1 and n y < n + 1, (n 0); f (x, y ) = an if n x < n + 1 and n + 1 y < n + 2, (n 0); while f (x, y ) = 0 elsewhere. Here an
Math 245A Homework 2
Brett Hemenway October 19, 2005
Real Analysis Gerald B. Folland Chapter 1. 13. Every -nite measure is seminite. Let be a -nite measure on X, M. Since is -nite, there exist a sequence of sets cfw_En in M, with (En ) < and En = X . n=1
A Hungerfords Algebra Solutions Manual
Volume I: Introduction through Chapter IV
James Wilson
D4
I
b 0 = C0 (G) C1 (G) Gn1 n G Local Ring
a2 , b a2 b
a a2 0
a2 , ab ab a3 b
Cn1 (G) Cn (G) = G G1 2 G G0 = G 1 G = G
II
0 = Gn 0 = n+1 G Commutative Ring
Fie
MATH 128A, SUMMER 2009: FINAL REVIEW PROBLEMS
(1) Find the largest n for which cos(x) = 1 x + O(xn ). Write the corresponding Taylor remainder. 2 (2) Match each of (1, 2, 3) below with the correct value (a, b, c). 1. Absolute error for the 2. Relative err
A The Comprehensive L TEX Symbol List Scott Pakin <scott+clsl@pakin.org> 22 September 2005
Abstract
A This document lists 3300 symbols and the corresponding L TEX commands that produce them. Some A of these symbols are guaranteed to be available in every
CalFIT
MON
7:00-7:55am CIRCUIT TRAINING
W eight Rm Annex - Amanda
Group X Class Schedule TUE WED THU
7:00-7:55am TAI CHI Combatives - Starfire 8:00-8:55am PILATES ON THE BALL Gold Gym - Coco 8:00-8:55am CARDIO DANCE Combatives - Damie 9:00-9:25am TONING E
X -pic Users Guide Y
Kristoer H. Rose krisrose@ens-lyon.fr
Version 3.7, February 16, 1999
Abstract X -pic is a package for typesetting graphs and diagrams Y using Knuths TEX typesetting system. X -pic works with Y most of the many formats available; e.g.,
Customizing lists with the enumitem package
Javier Bezos Version 2.2 2009-05-18
1
Introduction
A When I began to use L TEX several year ago, two particular points annoyed me because I found customizing them was very complicated headlines/footlines and lis
Adam Allan
Solutions to Atiyah Macdonald
1
Chapter 1 : Rings and Ideals
1.1. Show that the sum of a nilpotent element and a unit is a unit.
i If x is nilpotent, then 1 x is a unit with inverse i=0 x . So if u is a unit and x is nilpotent, then 1 1 v = 1 (
Homework and Exams
Assigned on Wednesday of each week and due the following Wednesday. You are encouraged to form study groups to toss around ideas, but each student is required to submit a write-up of the solutions in his/her own words. All homework assi
MATH 202A HOMEWORK 14 Exercise 1. (a) Proof. (i). R K (x)dx = R d (x/ )dx = R ( x )d x = R (x)dx = 1 (ii). R |K (x)|dx = R d |(x/ )|dx = R |( x )|d x = R |(x)|dx < since is integrable on R. Hence, |K (x)| is bounded. (iii). > 0, similarly, we know |x| |K
MATH 202A HOMEWORK 12 Exercise 4. Proof. Let g (x) =
b 0 b 0 b 0 b 0
cfw_xt |f (t)| dt. It is clear that the function in the integral is meat
b 0
surable and thus by Tonelli Theorem, we have cfw_xt |f (t)| dx dt = t
b |f (t)| t 0 b 0
g (x) =
b 0
cfw_xt
MATH 202A HOMEWORK 11 The following assignment is due Wed., Nov. 14. SS Ch.1, p. 38, Exercise # 2(b)(c)(d). SS Ch.1, p. 42, Exercise #18. SS Ch.1, p. 43, Exercise #22. SS Ch.1, p. 43, Exercise #23. SS Ch.1, p. 44-45, Exercise #32. Extra Problem. (a) Prove
Math 202: Topology and Measure Theory
Mark Borgschulte
PS10: Measurability, Limits, Borel-Cantelli Lemma
Professor: Justin Holmer November 7, 2007
Exercises (Stein and Shakarchi, Chapter 1, Ex. 5, 7, 8, 11, 16, 26)
5. Suppose E is any given set, and On is
Math 202A Homework 10
Roman Vaisberg
Problem 5. Suppose E is a given set, and On is the open set: On = cfw_x | d(x, E ) < 1/n. (a) Show if E is compact then, m(E ) = limn m(On ). Proof. Clearly we have Ok Ok+1 and E = On . Thus On E . Thus we would have t
Math 202A Homework 9
Roman Vaisberg
Problem 1. (a) Let A be an algebra of continuous real-valued function on a compact space X and assume that A separates the points of X . Prove that either A = C (X ) or there is a point p X such that f (p) = 0 for all f
MATH 202A HOMEWORK 9 The following assignment is due Wed., Oct. 31. Problem 1. (a) (Royden problem #43). Let A be an algebra of continuous realvalued functions on a compact space X and assume that A separates the points of X . Then either A = C (X ) or th
Math 202A Homework 8
Roman Vaisberg
Problem 1. (a) Let X be a Hausdor topological space for each I . Prove that if an innite number of the coordinate spaces X are non-compact, then each compact subset of the product X = I X is nowhere dense. Proof. Let K
MATH 202A HOMEWORK 8 Problem 1.(a) Proof. Suppose K is compact in I X . Since X is closed for all , I X is also Hausdor and thus K is closed. We just need to show that K = . Let x = (x ) K . By hint, is continuous, (K ) X is compact. Since the innite numb
MATH 202A HOMEWORK 8
The following assignment is due Wed., Oct. 24. Recall we dene a space X to be locally compact if for each x0 X , there is a neighborhood U of x0 such that U is compact. Problem 1. (a) Let X be a Hausdor topological space for each I .
Math 202A Homework 6
Roman Vaisberg
Problem 1. (a) Let X be a topological space, cfw_Cn be a collection of connected subsets such that any two of them have a point in common. Prove that G= Cn is connected. Suppose G is not connected, and let U and V be o
MATH 202A HOMEWORK 6 Problem 1. (a) Proof. G is a subspace of X . Suppose G is not connected, there is a separtaion. Hence, U1 , U2 X , s.t. (1) (2) (3) (4) G U1 U2 U1 G = U2 G = U1 U2 G =
By (2), 1 s.t. U1 C1 = . Hence, in C1 , C1 G U1 U2 ; U1 C1 = ; U1
MATH 202A HOMEWORK 6 This assignment is due Wed., Oct. 10. The topics addressed are connectedness, products and quotients. Problem 1. (a) Let X be a topological space, cfw_C be a collection of connected subsets such that any two of them have a point in c
Math 202a
Michael Phillips 9/26/07
Problem 1.(a.) Pick any x [0, 2 ]. sin kx has period 2k , for any > 0, we can take k large enough so that 2k < - i.e. so that at least two periods of sin kx wholly t within B (x, ). So, we would then have sin kx = sin k
MATH 202A HOMEWORK 4
This assignment is due Wed., Sept. 26. It deals with equicontinuity in C (X ); the concepts of base, subbase, and subspace for a topological space; and a study of the the lower limit topology on R. Let X be a compact metric space, and
MATH 202A HOMEWORK 3 This assignment is due Wednesday, Sept. 19, and deals primarily with compactness, the Baire Category theorem, and associated concepts. Problem 1. Prove that a metric space X is totally bounded if and only if every sequence in X has a
MATH 202A HOMEWORK 2 This assignment covers material from the second week of class. The corresponding reading material is Royden Ch 7, 3-7. Problem 1. Show that separability is a topological property, i.e. that it is preserved under homeomorphism. In the
MATH 202A HOMEWORK 1 SOLUTIONS Problem 1. Suppose p q . Let x = (xn ). We claim that (1) x
x
q
x
p q p
x
1 p q
The rst part of (1) is the observation that for xed j ,
|xj |
n=1
q
|xn |q = x
q
q
since |xj |q is one of the terms in the sum, and the remaini