AS/2401/3401/T1/2015-2016
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH2401/3401 Analysis I
Tutorial 1
Date: 9 Sep. 2016
1. Let S = cfw_(1)n + 1/n : n N R and T = cfw_(1)2n + 1/2n : n
N S.
(a) Is S open in R?
(b) Is S closed in R?
(c) Is T o
AS/2401/3401/2/2015-2016
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH2401/3401 Analysis I
Assignment 2
Due date: 30 Sep. 2015, before 5:00p.m.
1. For metric spaces (X, d) and S X, nd S , S and S where
(a) X = R, d is the usual Euclidean metr
AS/2401/3401/1/2015-2016
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH2401/3401 Analysis I
Assignment 1
Due date: 14 Sep. 2015, before 5:00p.m.
1. Let d1 , d2 : Rn Rn R as
d1 (x, y) = max |xi yi |
1in
d2 (x, y) =
n
|xi yi |
i=1
for x = (x1 ,
LN-S/Analysis I/WSC/MS/2013-2014
Department of Mathematics
The University of Hong Kong
MATH2401 Analysis I
Chapter 1:
Metric Spaces
1.1 Denitions and Examples
Denition. Let X be a non-empty set. A metric on X is a real-valued function
d:X X R
satisfying
(
LN-S/Analysis I/WSC/MS/2013-2014
Chapter 2:
Limits and Continuity
2.1 Convergence in a Metric Space
Denition. A sequence cfw_xn in X is said to be convergent and converge to a X if for every > 0,
there exists N N such that
d(xn , a) < whenever n N .
In t
LN-S/Analysis I/WSC/MS/2013-2014
Chapter 3:
Connectedness
3.1 Connectedness
Denition. A metric space X is said to be disconnected if X = A B, where A = , B = , A B = ,
and both A, B are open in X. X is said to be connected if it is not disconnected. A sub
LN-S/Analysis I/WSC/MS/2013-2014
Chapter 4:
Uniform Continuity and Uniform Convergence
4.1 Uniform Continuity and Banachs Fixed Point Theorem
Denition. A function f : X Y is said to be uniformly continuous on a subset S X if for any
> 0, there exists = (
AS/2401/3401/1/2013-2014
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH2401/3401 Analysis I
Assignment 1
Due date: 16 Sep. 2013, before 5:00p.m.
1. Let X = R2 and for all x = (x1 , x2 ), y = (y1 , y2 ) X, dene
d(x, y) =
|x1 y1 |
|x1 | + |x2 y2
MATH2401 / MATH3401 Analysis I
Quiz 2 (Nov 11, 2013)
Solutions
(
1. a) True.
a A, a sequence cfw_xn in A. s.t. xn a. Hence cfw_xn is Cauchy. Since A is complete,
xn x A. Hence a = x A.
(b) True.
f 1 (T ) is open and f 1 (T ) f 1 (T ). Thus f 1 (T ) f 1
Probability Theory
MATH 2603/3603
Zheng Qu
The University of Hong Kong
Lecture 5
September 18, 2017
1 / 14
Previously
I
Independent Random Variables
I
Covariance
I
Moment Generating Function
I
Markovs Inequality
I
Chebyshevs Inequality
I
Weak/Strong Law o
AS/2401/3401/3/2015-2016
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH2401/3401 Analysis I
Assignment 3
Due date: 26 Oct. 2015, before 5:00p.m.
1. Let S X be compact.
(a) Show that every sequence of S has at least one convergent subsequence.
AS/2401/3401/2/2015-2016
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH2401/3401 Analysis I
Assignment 4
Due date: 9 Nov. 2015, before 5:00p.m.
1. Let A R be a nonempty and connected set. Assume that A Q.
Show that A is a singleton.
2. Let X b
LN-S/Analysis I/2015-2016
Department of Mathematics
The University of Hong Kong
MATH2401/3401 Analysis I
Chapter 1:
Metric Spaces
1.1 Denitions and Examples
Denition. Let X be a non-empty set. A metric on X is a real-valued function
d:X X R
satisfying
(M1
AS/2401/3401/T1/2015-2016
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH2401/3401 Analysis I
Tutorial 1
Date: 9 Sep. 2016
1. Let S = cfw_(1)n + 1/n : n N R and T = cfw_(1)2n + 1/2n : n
N S.
(a) Is S open in R?
(b) Is S closed in R?
(c) Is T o
AS/2401/3401/1/2016-2017
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH2401/3401 Analysis I
Assignment 1
Due date: 12 Sep. 2016, before 5:00p.m.
1. Prove or disprove that (X, d) is a metric space where
(a) X = R, d(x, y) = x2 y 2 ;
(b) X = R,
AS/2401/3401/2/2016-2017
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH2401/3401 Analysis I
Assignment 2
Due date: 26 Sep. 2016, before 5:00p.m.
1. Let (X, d) be a metric space and S, T be subsets of X. Prove or
disprove (with counterexample)
AS/2401/3401/1/2016-2017
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH2401/3401 Analysis I
Assignment 1
Due date: 12 Sep. 2016, before 5:00p.m.
1. Prove or disprove that (X, d) is a metric space where
(a) X = R, d(x, y) = x2 y 2 ;
(b) X = R,
AS/2401/3401/T1/2015-2016
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH2401/3401 Analysis I
Tutorial 1
Date: 11 Sep. 2015
1. Let S = cfw_(1)n + 1/n : n N R and T = cfw_(1)2n + 1/2n : n
N S.
(a) Is S open in R?
(b) Is S closed in R?
(c) Is T
AS/2401/3401/T2/2015-2016
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH2401/3401 Analysis I
Tutorial 2
Date: 2 Oct. 2015
1. Let (X, d) be a metric space and S X. Show that (under the following settings) S is closed and bounded and determine w
MATH2401 / MATH3401 Analysis I
Solutions (Test 1 Oct 8, 2015)
(1) False.
Example: X := (0, 1) cfw_2 R, S := cfw_2.
(2) True.
Write S = cfw_x1 , . . . , xn . For any open cover cfw_U of S, i, i s.t. xi Ui .
Then cfw_Ui n is a nite subcover.
i=1
(3) False.
LN-S/Analysis I/2015-2016
Chapter 2:
Limits and Continuity
2.1 Convergence in a Metric Space
Denition. A sequence cfw_xn in X is said to be convergent and converge to a X if for every > 0,
there exists N N such that
d(xn , a) < whenever n N .
In this cas
LN-S/Analysis I/2015-2016
Chapter 3:
Connectedness
3.1 Connectedness
Denition. A metric space X is said to be disconnected if X = A B, where A = , B = , A B = ,
and both A, B are open in X. X is said to be connected if it is not disconnected. A subset S X
Probability Theory
MATH 2603/3603
Zheng Qu
The University of Hong Kong
Lecture 4
September 14, 2017
1 / 15
Previously
I
Joint Cumulative Distribution Function / Joint Probability Mass
Function/Joint Probability Density Function
2 / 15
Today
I
Independent
AS/2401/3401/T2/sol/2013-2014
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH2401/3401 Analysis I
Solution to Tutorial 2
Date: 03 Oct. 2013
1. (a) Note that
r > 0 (B(x, r) A = )
r > 0 [ar A (d(ar , x) < r)]
r > 0 (d(x, A) < r)
xA
d(x, A) = 0.
(
AS/2401/3401/T2/2013-2014
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH2401/3401 Analysis I
Tutorial 2
Date: 3 Oct. 2013
1. Let (X, d) be a metric space, x X and ( =)A X. Recall that the
distance between x and A is dened by d(x, A) = inf d(x,
AS/2401/3401/5/2013-2014
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH2401/3401 Analysis I
Assignment 5
Due date: 18 Nov. 2013, before 5:00p.m.
1. Let n > 1 and B Rn be countable. Show that Rn \ B is path
connected.
(Hint is available in MOOD
AS/2401/3401/2/2013-2014
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH2401/3401 Analysis I
Assignment 2
Due date: 30 Sep. 2013, before 5:00p.m.
1. Let (X, d) be a metric space and S X. Show that
(a) (S ) S .
(b) S = S X \ S and S = (X \ S).
2
AS/2401/3401/4/2013-2014
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH2401/3401 Analysis I
Assignment 4
Due date: 4 Nov. 2013, before 5:00p.m.
Convention and Facts
Throughout, (X, dX ) and (Y, dY ) are nonempty metric spaces unless specied ot
AS/2401/3401/Sol/3/2013-2014
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH2401/3401 Analysis I
Outlined Solution to Assignment 3
1. Let S be the image of the sequence, that is,
S = cfw_x X : x = xn for some n .
If S is nite, then the pre-ima
AS/2401/3401/Sol/1/2013-2014
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH2401/3401 Analysis I
Outlined Solution to Assignment 1
1. Yes. (M1) and (M2) clearly hold. We only verify (M3). Let x =
(x1 , x2 ), y = (y1 , y2 ) and z = (z1 , z2 ) be
Math 132 - Topology II: Smooth Manifolds. Spring 2017.
Homework 9
Due in class by 1.15pm, Friday April 7.
Late homework will not be accepted.
Note: A.x.y refers to Problem y in Section A.x in Differential Topology by Guillemin &
Pollack, AMS Chelsea Pub.
Math 132 - Topology II: Smooth Manifolds. Spring 2017.
Homework 8
Due in class by 1.15pm, Friday March 31.
Late homework will not be accepted.
Note: A.x.y refers to Problem y in Section A.x in Differential Topology by Guillemin &
Pollack, AMS Chelsea Pub.
Math 132 - Topology II: Smooth Manifolds. Spring 2017.
Homework 6
Due in class by 1.15pm, Wednesday March 8.
Late homework will not be accepted.
Note: A.x.y refers to Problem y in Section A.x in Differential Topology by Guillemin &
Pollack, AMS Chelsea Pu
Math 132 - Topology II: Smooth Manifolds. Spring 2017.
Homework 7
Due in class by 1.15pm, Monday March 27.
Late homework will not be accepted.
Note: A.x.y refers to Problem y in Section A.x in Differential Topology by Guillemin &
Pollack, AMS Chelsea Pub.