1
Chester Pak Hei Cheng
Functional analysis
Summary of research:
My research interest is in the field of functional analysis, with focus on operator theory. Functional
analysis is a place where one studies the topological effects on linear maps between ve

PMath 351 Notes
Laurent W. Marcoux
Department of Pure Mathematics
University of Waterloo
Waterloo, Ontario Canada N2L 3G1
August 8, 2014
Preface to the First Edition - May 6, 2014
The following is a set of class notes for the PMath 351 course I am current

Assignment I
PMATH 351
Due: May 26 2015, 8:30
Question 1 For each of the following subsets of R R determine whether it is equal to the
Cartesian product of two subsets of R. If the answer is negative justify your answer.
1. X = cfw_(x, y)|x is an integer.

Assignment II
PMATH 351
Due: June 2, 2015, at 8:30
Question 1 (Lemma 5.3)
Let , , be cardinal numbers. Prove that.
(a) The sum + is well-defined, that is if = |A| = |C| and = |B| = |D| for some
A B = and C D = , then |A B| = |C D|.
(b) = and () = ().
(c)

Assignment III (Solution)
PMATH 351
Due: June 19, 2015, at 8:30
Question 1 Let d1 , d2 and d be the metrics on Rn given by
n
X
d1 (x, y) =
|xi yi |
i=1
v
u n
uX
|xi yi |2
dn (x, y) = t
i=1
and
n
d (x, y) = sup |xi yi |.
i=1
Let 1 , 2 and be the topologies

Assignment II
PMATH 351
Due: June 2, 2015, at 8:30
Question 1 (Lemma 5.3)
Let , , be cardinal numbers. Prove that.
(a) The sum + is well-defined, that is if = |A| = |C| and = |B| = |D| for some
A B = and C D = , then |A B| = |C D|.
(b) = and () = ().
(c)

Solution Sheet 1
1.
(i) Let 1 < p < and
ap
p
+
0
bp
p0
1
p
+
1
p0
= 1. By finding the minimum of f (b) =
ab, or otherwise, prove Youngs inequality: For all a, b > 0,
0
ap b p
ab
+ 0,
p
p
(ii) Deduce Youngs inequality with : For all a, b > 0, > 0, p as a

Assignment III
PMATH 351
Due: June 19, 2015, at 8:30
Question 1 Let d1 , d2 and d be the metrics on Rn given by
d1 (x, y) =
n
X
|xi yi |
i=1
v
u n
uX
|xi yi |2
dn (x, y) = t
i=1
and
n
d (x, y) = sup |xi yi |.
i=1
Let 1 , 2 and be the topologies induced by

CHAPTER 1
Metric Spaces
1. Definition and examples
Metric spaces generalize and clarify the notion of distance in the real line. The
definitions will provide us with a useful tool for more general applications of the
notion of distance:
Definition 1.1. A

Assignment III
PMATH 351
Due: June 19, 2015, at 8:30
Question 1 Let d1 , d2 and d be the metrics on Rn given by
d1 (x, y) =
n
X
|xi yi |
i=1
v
u n
uX
dn (x, y) = t
|xi yi |2
i=1
and
n
d (x, y) = sup |xi yi |.
i=1
Let 1 , 2 and be the topologies induced by

Math 138 - Power series and Taylor polynomial
Power series
Taylor series is a particular example of a large class of functions called the power series
f (x) =
X
an x n
n=0
Maclaurins idea was to try making a polynomial that agrees with some function f (x)

Math 138 - Sequence
At its most abstract, a sequence is just a collection of numbers.
Notation:
[xn ]
n=1 = [x1 , x2 , ., xn ]
A sequence [xn ] has a limit, p, provided xn is as close as p as we like (That is for all epsilon 0) by
simply taking n large en

1
Phys 476
Assignment 3
Chester Cheng id:20488508
Problem 1:
a.)
Just like the skew symmetric case, the diagonals are zero so there are
nents
b.)
N 3 (N 1)
independent compo2
N 3 (N 1)
2
c.)
Since the last two indices also have skew symmetric property, we

Math 138 - Series
Inifinite series
The connection between series and sequence is through the partial sum of the series.
Sn =
N
X
Cn
n=0
Does the sequence converge as N ? If it does, then Cn converges. To verify if a series converges,
we will deal with lot

1
Chester Cheng id:20488508 Assignment 1 Pmath 348
Question 1.)
Using Gausss lemma, we can reduce the problem to Z[x] by first making the polynomial f (x)
primitive, which is always possible since we can divide by gcds.
Suppose for a contradiction that f

Chapter 3
METRIC SPACES and SOME
BASIC TOPOLOGY
Thus far, our focus has been on studying, reviewing, and/or developing an understanding and ability to make use of properties of U U1 . The next goal is to
generalize our work to Un and, eventually, to study