23/2/04
MA3056 Exercise Sheet 3: Compactness, Connectedness
1. Show that every nite topological space is compact. Show that a discrete topological
space is compact if and only if it is nite.
2. Let Y be a subspace of a topological space X and let Z be a n
201A, Fall 10, Thomases
Homework 5 Solutions
Mihaela Ifrim
1. If f is continuous on [0, 1] and if
1
f (x)xn dx = 0,
n N ,
0
then f (x) = 0 on [0, 1].
Hint. The integral of the product of f with any polynomial is zero.
1
Use the Weierstrass theorem to show
Math 320 Assignment 09, out of 25
send corrections to db5dmath.ubc.ca
In proofs it is a good habit to say when a hypothesis is being used.
1. (5 points) 4 #3. Let f be a continuous real function on a metric space X. Let Z(f ) be the set of all
p X at whic
201A nal practice solutions - Thomases
1. Let f : [0, 1] (0, 1) be a function of class C 1 - that is to say both f
and f are continuous functions. Suppose further that
max |f (x)| 1
0x1
for some
> 0.
(a) Show that f has exactly one xed point.
(b) Let x0
201A, Fall 10, Thomases
Homework 4 - Solutions
Mihaela Ifrim
1. Suppose fn C ([0, 1]) is a monotone decreasing sequence that converges pointwise to f C ([0, 1]). Prove that fn converges uniformly
to f . This result is called Dini s monotone convergence th
201A, Fall 10, Thomases
Homework 3
Mihaela Ifrim
1. Let X be a normed linear space. A series in
xn in X is absolutely
convergent if
xn converges to a nite value in R. Prove X is a
Banach space if and only if every absolutely convergent series converges.
P
MA2223: SOLUTIONS TO PROBLEM SHEET 4
1. Let A and B be open sets in a normed vector space (X, . ) and let
r be a positive real number. Prove that the following sets are open
(a)
A + B = cfw_a + b : a A, b B
(b)
rA = cfw_ra : a A
Solution:
(a) We can expre
MA2223: SOLUTIONS TO PROBLEM SHEET 2
1. Let (X, d) be a metric space and let A be a subset of X.
(a) Prove that the closure A is the intersection of all closed sets
which contain A.
(b) Using (a) show that the closure of A is closed.
Solution: (a) Suppose
MA2223: SOLUTIONS TO PROBLEM SHEET 1
1. Show that the diameter of an open ball B(a, r) in Euclidean space
Rn is exactly 2r.
Solution: We proved in class that in any metric space the diameter of an open ball is at most twice the radius. So we know
diam(B(a
201A, Fall 10, Thomases
Homework 7
Mihaela Ifrim
1. Suppose cfw_xn is a weakly convergent sequence in a Banach space X . Show
that the weak limit of cfw_xn is unique.
Proof. Suppose cfw_xn converges weakly to x and y in X . Then for any
X , we nd
lim (xn
201A, Fall 10, Thomases
Mihaela Ifrim
Homework 6
Due on November 10th, 2010
1. For any f C ([0, 1]), we dene
1/2
1
f
1
2
|f (x)| dx
=
0
and
1/2
1
f
2
(1 + x)|f (x)|2 dx
=
.
0
Show that
1
and
2
are equivalent norms in C ([0, 1]).
Proof.
Let w(x) > 0 be
Math 421, Homework #8 Solutions
(1) Find an example of a function f : R2 \ cfw_0 R for which lim(x,y)0 f (x, y ) exists, but the iterated
limits limx0 limy0 f (x, y ) and limy0 limx0 f (x, y ) do not exist.
Answer. Note that many correct answers are possi
Math 421, Homework #9 Solutions
(1) (a) A set E Rn is said to be path connected if for any pair of points x E and y E there
exists a continuous function : [0, 1] Rn satisfying (0) = x, (1) = y, and (t) E for all
t [0, 1]. Let E Rn and assume that E is pat
Homework 8 Solutions
Math 171, Spring 2010
Henry Adams
44.2. (a) Prove that f (x) = x is uniformly continuous on [0, ).
(b) Prove that f (x) = x3 is not uniformly continuous on R.
Solution. (a) Given > 0, pick = 2 . First note that | x y| | x + y|. Hence
Math 421, Homework #7 Solutions
(1) Let cfw_xk and cfw_yk be convergent sequences in Rn , and assume that limk xk = L and that
limk yk = M. Prove directly from denition 9.1 (i.e. dont use Theorem 9.2) that:
(a) limk xk + yk = L + M.
Proof. Let > 0. The
Math 421, Homework #6 Solutions
(1) Let E Rn Show that
o
c
E = (E c ) ,
i.e. the complement of the closure is the interior of the complement.1
Proof. Before giving the proof we recall characterizations of the interior and closure (proved in
lecture) that
Math 421, Homework #5 Solutions
(1) (8.3.6) Suppose that E Rn and C is a subset of E .
(a) Prove that if E is closed, then C is relatively closed in E if and only if C is a closed set (as
dened in Denition 8.20(ii).
Proof. First assume that C is a closed
Math 421, Homework #4 Solutions
Note: In problems (1) and (2) we will denote the operator norm by L to distinguish it from the usual
norm on Rn .
(1) (8.2.11) Let T L(Rn , Rm ), and dene
M1 := sup
T(x) = sup
x Rn ; x = 1
T(x)
x =1
T(x) C x for all x Rn .
201A Midterm - November 1, 2010 - Thomases
Name:
Show all of your work, in particular note any theorems you may use and
state the conditions carefully and make sure they are satised.
Problem Possible Points Points Received
1
10
2
10
3
10
4
10
5
10
Total
5
MA2223: SOLUTIONS TO ASSIGNMENT 4
1. Prove directly that the following three norms on R2 are equivalent.
(a) x
1
= |x1 | + |x2 |
(b) x
2
=
(c) x
(x1 )2 + (x2 )2
= maxi=1,2 |xi |
where x = (x1 , x2 ).
Solution: First we will show that the 1-norm and the ma
Real Analysis Homework 2 Solutions
7. Let X be a metric space and cfw_Fi iI be a collection of closed subsets of X. Let F = Fi .
iI
We will give two proofs that F is closed. Proof 1. Let x be a cluster point of F . We will show that x F . Let U be any nei
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METRIC AND TOPOLOGICAL SPACES
ALEX GONZALEZ
1. Introduction
2. Metric spaces: basic denitions
2.1. Normed real vector spaces
2.2. Product spaces
3. Open subsets
3.1. Equivalent metrics
3.2. Properties of open subsets and a bit of set theory
3.3. Convergen
6/2/04
MA3056 Exercise Sheet 1: Solutions and Hints
1
1. (b) Solution: If f (t) = 0 for all t then clearly 0 f (t) dt = 0, so suppose that f is a
nonzero function. Hence f (t0 ) > 0 for some point t0 (0, 1). By continuity of f there
is some > 0 such that
Topology I - Exercises and Solutions
July 25, 2014
1
Metric spaces
1. Let p be a prime number, and d : Z Z [0, +) be a function dened by
dp (x, y) = p maxcfw_mN
: pm |xy
.
Prove that dp is a metric on Z and that dp (x, y) maxcfw_dp (x, z), dp (z, y) for e
24/2/04
MA3056 Exercise Sheet 3: Solutions and Hints
2. Solution: Let T denote the topology on X and TY the subspace topology on Y .
Suppose that Z is compact as a subspace of Y , and let cfw_U T be a collection of
open subsets of X such that Z U . Since
221: Solutions to Assignments
Assignment 1
1. Let (X, d) be a metric space.
Show that d3 : X X R is a metric for X where
d3 (x, y) =
d(x, y)
for all x, y X.
1 + d(x, y)
Since d is a metric on X we have for all x, y X:
d3 (x, y) =
d3 (x, y) =
d(x, y)
0
1 +
2S1: Probability and Statistics
Topics covered in this section:
Review of elementary probability and statistics
Discrete random variables
Binomial distribution, Poisson distribution
Continuous random variables
Normal distribution, central limit theor
Homework 5 Solutions
Math 171, Spring 2010
Henry Adams
29.15. Let a1 = 1 and an = 1/2n1 for n 2. Let cfw_bm,n
a1 a1 a1 a1
0 a2 a2 a2
0 0 a3 a3
0 0 0 a4
Show that the sum by columns of the series
m,n bm,n
be the double sequence whose matrix is
.
. . .
.