Solution exercises of Rudin  Functional Analysis Book
TNH 001

Fall 2016
Notes on
Partial Differential Equations
John K. Hunter
Department of Mathematics, University of California at Davis1
1
Revised 6/18/2014. Thanks to Kris Jenssen and Jan Koch for corrections. Supported in
part by NSF Grant #DMS1312342.
Abstract. These are
Solution exercises of Rudin  Functional Analysis Book
TNH 001

Fall 2016
Inequalities From Around the World 19952005
Solutions to Inequalities through problems by Hojoo Lee
Autors: Mathlink Members
Editor: Ercole Suppa
Teramo, 28 March 2011  Version 1
I
Introduction
The aim of this work is to provide solutions to problems on
Solution exercises of Rudin  Functional Analysis Book
TNH 001

Fall 2016
Bt ng thc Schur v phng php i
bin p,q,r
V Thnh Vn
Lp 11 TonKhi chuyn THPTHKH Hu
Nh cc bn bit, bt ng thc Schur l mt bt ng thc mnh v c nhiu ng dng, tuy nhin n vn
cn kh xa l vi nhiu bn hc sinh THCS cng nh THPT. Qua b i vit n y, ti mun cng cp thm cho
cc bn m
Solution exercises of Rudin  Functional Analysis Book
TNH 001

Fall 2016
Functional Equations Problems
Amir Hossein Parvardi
June 13, 2011
Dedicated to pco.
email:
ahpwsog@gmail.com, blog: http:/matholympiad.blogsky.com.
1
1
Definitions
N is the set of positive integers.
N cfw_0 = N is the set of nonnegative integers.
Z
Solution exercises of Rudin  Functional Analysis Book
TNH 001

Fall 2016
1
Vietnam Inequality Forum
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Solution exercises of Rudin  Functional Analysis Book
TNH 001

Fall 2016
Problem Set 4. Due in class Thursday, November 3.
Math 322, Fall 2016
(1) Let G be a cyclic group of order pqr, where p, q and r are distinct primes. How
many subgroups does G have (counting the trivial subgroup cfw_e and G itself).
(2) Let G be a finite
Solution exercises of Rudin  Functional Analysis Book
TNH 001

Fall 2016
Math 417 C1
HOUR EXAM II
18 July 2005
SOLUTIONS
1. Show that for every subgroup H of the symmetric group Sn , either all of the
elements are even or exactly half of them are even.
SOLUTION: Since (1) is even, H has some even elements. If all elements of H
Solution exercises of Rudin  Functional Analysis Book
TNH 001

Fall 2016
Math 322: Problem Set 6 (due 27/10/2015)
Practice problems
P1. Let G be a group and let X be a set of size at least 2. Fix x0 X and for g G, x X set
g x = x0 .
(a) Show that this operation satisfies (gh) x = g (h x) for all g, h G, x X.
(b) This is not a
Solution exercises of Rudin  Functional Analysis Book
TNH 001

Fall 2016
Problem Set 6. Due in class Thursday, December 1.
Math 322, Fall 2016
(1) Suppose a pgroup G acts on a finite set X. If this action has n fixed points in X,
show that n X (mod p).
(Recall that x X is called a fixed point if g x = x for every g G.)
(2)
Solution exercises of Rudin  Functional Analysis Book
TNH 001

Fall 2016
TOPOLOGICAL VECTOR SPACES 37
(b) If X is locally convex then the convex hull of every bounded set is bounded. (This is
false without local convexity; see Section 1.47.)
(c) If A and B are bounded, so is A + B.
(d) If A and B are compact, so is A + B.
(e)
Solution exercises of Rudin  Functional Analysis Book
TNH 001

Fall 2016
3.
(a) We have already known A is an open subset of X. Since co(A) is open
and co(A) is convex, co(A) is also convex. Meanwhile, co(A) is the minimum
convex set containing A, so co(A) co(A) co(A). Therefore, co(A) =
co(A) , i.e. co(A) is open.
(b) Since X
Solution exercises of Rudin  Functional Analysis Book
TNH 001

Fall 2016
By Thanh Nhan
P38
1.
(a) If z1 , z2 , such that x + z1 = y, x + z2 = y, then x + z1 = x + z2 , we
can get that x + x + z1 = x + x + z2 . Hence z1 = z2 .
(b) (0 + 0)x = 0x + 0x = 0x 0x = 0
0 = (0 + 0) = 0 + 0 0 = 0
(c) x 2A, a A, such that x = 2a = a + a,