PROBLEM SET NO. 1
Carothers Ch. 1: Exercises 15, 21, 26, 41
Carothers Ch. 2: Exercises 5, 16
Problem set due on Thursday, 10/10, 4:00 PM, in math departments drop box
1
Math 113 (Spring 2009) Yum-Tong Siu Solutions of Problems of Math 113 Final Examination May 20, 2009, 9:15 a.m. - 12:15 p.m. Science Center A Total Number of Points = 200
1
Problem 1. Evaluate the following two denite integrals by using the theory of resi
HOMEWORK 8 FOR MA108A
DUE DATE: 4PM, THURSDAY DECEMBER 2, 2010
1. Homework
4
2
(1) Let f : 2 R be given by f (y ) = n (yn + yn /n). f is well dened (i.e. the series converges), as
we know that elements of 2 are also elements of 4 , and the functions is bo
COMPLETING METRIC SPACES USING CAUCHY-SEQUENCES
FOKKO VAN DE BULT
Suppose (M, d) is a metric space. We want to show it has a completion (we
do not concern ourselves with uniqueness in this note). The main idea is to add
points which can serve as limits fo
PROBLEM SET NO. 5
Carothers Ch. 7: Exercises 42
Carothers Ch. 8: Exercises 2, 10, 13, 35, 70
Problem set due on Thursday, 11/14, 4:00 PM, in math departments drop box
1
HOMEWORK NO.4
MA108A FALL 2013
(1) Problem 40 of Chapter 5
Solution: Dene
A := cfw_(x, y ) : x 0; y = ex and B := cfw_(x, y ) : x 0; y = 0.
Then both A and B are nonempty closed sets, and A B = .
Moreover, for any > 0, choose x0 = log + 1. Then
d(A, B )
HOMEWORK NO.2
MA108A FALL 2013
(1) Problem 26 of Chapter 2
P
P
n
Proof Let x = 1 23xnn and y = 1 23yn . Suppose f (x) >
n=1 P
n=1
P 1 xn
1 yn
f (y ). Then n=1 2n > n=1 2n . Let kP mincfw_i : xi > yi .
=
P 1 yn
1 xn
Then k < 1 since otherwise we have
n=1 2
MA108 HOMEWORK 3 SOLUTIONS
1. Problem 1
Let y Ac , then d( x, y) > r and thus d( x, y) > r + for some
that B (y) Ac and so Ac is open and thus A is closed as desired.
> 0. It is then follows
Let M be a set with more than one point. Dene a function d on M
PROBLEM SET NO. 4
Carothers Ch. 5: Exercises 40, 50
Carothers Ch. 6: Exercises 2, 6, 9, 12
Problem set due on Thursday, 10/31, 4:00 PM, in math departments drop box
1
MIDTERM FOR MA108A DUE DATE: 4PM, TUESDAY NOVEMBER 2, 2010
Read the instructions on the separate sheet before you begin! (1) Let C 1 [0, 1] be the vectorspace of all continuously differentiable functions on the interval [0, 1]. We know that f = supx[0,1]
FINAL FOR MA108A
DUE DATE: 4PM, FRIDAY DECEMBER 10, 2010
Read the instructions on the separate sheet before you begin!
(1) Suppose f : X Y is a continuous map from a compact metric space X to an arbitrary metric
space Y .
(a) Show that for any A X we have
HOMEWORK 3 FOR MA108A
DUE DATE: 4PM, TUESDAY OCTOBER 19, 2010
1. Homework
(1) Let G be open and D dense in M . Show that G D is dense in G. Show that this is not necessarily
true if G is not open.
Proof. To show that G D is dense in G, it suces to show th
HOMEWORK 5 FOR MA108A
DUE DATE: 4PM, TUESDAY NOVEMBER 9, 2010
1. Homework
(1) Let 1 p < . Let c R . Show that Sc = cfw_x
only if c p .
What goes wrong if we take p = ?
p
| |xn | cn is a compact subset of
p
if and
Proof. First we show that if 1 p < and c
HOMEWORK 7 FOR MA108A
DUE DATE: 4PM, TUESDAY NOVEMBER 23, 2010
1. Homework
pw
(1) (a) Let fn f , where fn : X Y , with X a compact metric space. Suppose fn forms an
uc
equicontinuous sequence (i.e. cfw_fn is equicontinuous). Show that fn f .
(b) X compac
HOMEWORK 4 FOR MA108A
DUE DATE: 4PM, TUESDAY OCTOBER 26, 2010
1. Homework
(1) (Combination of 5.46 and 5.51) Let H be the Hilbert cube, dened as the set of sequences x
with x 1, with metric
2n |xn yn |
dH (x, y ) =
n=1
(This is a metric by Set 2 problem
HOMEWORK 6 FOR MA108A
DUE DATE: 4PM, TUESDAY NOVEMBER 16, 2010
1. Homework
uc
(1) Suppose fn : M N is continuous for all n (with M and N metric spaces). Suppose fn f .
Suppose xn M such that xn x. Show that fn (xn ) f (x).
Proof. Let us rst remark that f
MA 108A, HOMEWORK 3 SOLUTIONS
11. dim Mn,m (R) = mn. The standard basis for M is the collection of
mn matrices Eij which have 1 in the position (i, j ) and 0 everywhere
else. Let us show that A is a norm. Clearly A 0 for all A
and O = 0 (O is the zero mat
MA 108A, HOMEWORK 4 SOLUTIONS
16. Assume f has no xed points. Then g (x) = |f (x) x| > 0 on K
and so attains a strictly positive minimum value at some x0 K . But
in this case f (x0 ) = x0 and so g (f (x0 ) < g (x0 ), i.e. g (x0 ) is not a
minimum. Thus a
MA 108A, HOMEWORK 6 SOLUTIONS
26. Dh = (hu , hv ), where hu =
g (f ), fu and hv =
g (f ), fv .
27. f (x) is dierentiable at every point. For x = 0 we have fx (x, y ) =
4 1/ 3
x sin(y/x) x2/3 y cos(y/x) and fy (x, y ) = x1/3 cos(y/x). Thus the
3
partial de
HOMEWORK 2 FOR MA108A
DUE DATE: 4PM, TUESDAY OCTOBER 12, 2010
1. Homework
(1) Let f : R0 R0 be a concave strictly increasing function with f (0) = 0. Moreover let an denote
a sequence of positive real numbers. Dene a function d : R R R cfw_ as
ak f (|xk y
HOMEWORK 1 FOR MA108A
DUE DATE: 4PM, TUESDAY OCTOBER 6, 2009
1. Homework
n
(1) (a) Consider the sequence dened recursively by x1 = 1 and xn+1 = x2 + x1 . Show that xn
n
converges and determine its limit.
(b) Given two positive numbers x and y , such that