1.1
Class Notes for PMath 351, Spring Term 2012 Lecture 1: Metric spaces definition, examples. Open and closed balls in a metric space
A. Definition of a metric space, examples
1.1 Definition. 1o Let X be a non-empty set. A distance function on X is a fun

2.1
Class Notes for PMath 351, Spring Term 2012 Lecture 2: Convergent sequences in a metric space
2.1 Notation. Let (X, d) be a metric space. We will use the notation "(xn ) " for a n=1 sequence x1 , x2 , . . . , xn , . . . of elements of X. [Note: A sequ

3.1
Class Notes for PMath 351, Spring Term 2012 Lecture 3: Cluster points, Cauchy sequences. The completeness property for a metric space
A. Cluster points 3.1 Definition. Let (X, d) be a metric space, let (xn ) be a sequence of elements of n=1 X, and let

4.1
Class Notes for PMath 351, Spring Term 2012
Lecture 4: Two properties for sets in a metric space:
separable and totally bounded
In Lecture 1 we saw the concept of bounded set in a metric space (X, d), and in
connection to that we saw what it means for

6.1
Class Notes for PMath 351, Spring Term 2012 Lecture 6: Objects = metric spaces. What are the "morphisms"?
In this lecture we start to look at functions between metric spaces, and we address the natural question of what are the "right" kind of function

7.1
Class Notes for PMath 351, Spring Term 2012
Lecture 7: Objects = normed vector spaces. What are the morphisms?
This is a little spin-o of the preceding lecture, where we only focus on normed vector
spaces. The preferred functions between two vector sp

PMATH 351, SPRING TERM 2012 (INSTRUCTOR ALEXANDRU NICA) PART 5 OF THE COURSE, LECTURES 2227: SPACES OF CONTINUOUS FUNCTIONS
22.1
Class Notes for PMath 351, Spring Term 2012 Lecture 22: Pointwise and uniform convergence for a sequence of functions. The Ban

1
PMath 351, Spring Term 2012
This is concerning the posting of Class Notes on the D2L web-site of the course. The
notes are for Lectures 0-27, and are divided into 5 parts.
At the present moment, Class Notes have been posted in advance for the whole term

1
PMath 351, Spring Term 2012 A quick readiness test
PMath 351 is, first and foremost, a course of analysis on metric spaces. The framework of a metric space axiomatizes the concept of "distance between two points in the space". It is a very simple and na

1
PMath 351, Spring Term 2012
Solutions to homework assignment 3
Problem P2.5 Let (X, d) be a metric space, let a be an isolated point of X , and let
(xn ) be a sequence in X . Prove the following equivalence:
n=1
(xn ) converges
n=1
to a in (X, d)
there

1
PMath 351, Spring Term 2012
Solutions to homework assignment 4
Problem P2.9 (any distance is equivalent to a bounded distance).
Let (X, d) be a metric space. Dene d : X X R by the following formula
d(x, y ) =
d(x, y ), if d(x, y ) 1
1,
if d(x, y ) > 1
,

1
PMath 351, Spring Term 2012 Bonus homework assignment 10 Solutions
Theorem E (extension). Let (X, d) and (Y, d ) be metric spaces, where it is given that (Y, d ) is complete. Let A be a subset of X which is dense in X (we have cl(A) = X). We view A as a

1
PMath 351, Spring Term 2012
Solutions to Homework Assignment 7
Problem P13.3 Let (X, d) be a complete metric space, and let
B (x1 ; r1 ) B (x2 ; r2 ) B (xn ; rn )
be a nested sequence of closed balls in X , such that r1 > r2 > > rn > and
limn rn = 1. S

1
PMath 351, Spring Term 2012 Homework assignment 1 Solutions
Problem P0.4 Let A and B be two countable sets. Prove that the Cartesian product A B := cfw_(a, b) | a A, b B is countable. Solution. If A = or B = then A B = and is therefore countable. In wha

PMath 351, Spring Term 2012
Information on the Final Exam
The nal exam of PMath 351 is scheduled on
Tuesday, August 7, from 4 to 6:30 pm.
The exam will be written in RCH 211.
Types of questions. The questions on the nal exam will be of the same
three type

1
PMath 351, Spring Term 2012
This is a note concerning the classroom where the course is held. On
Monday, May 28 and Wednesday, May 30
we were assigned a dierent classroom than the usual one; it is
MC 5136B,
on the 5th oor of the MC building. Please make