Analysis III Mat 3120
Wednesday, May 21, 2014
Assignment 3
(given Wednesday, May 21, due Wednesday, May 28, at the beginning of the lecture)
(1) Sometimes one says that a metric space X is dense in itself if it contains no
isolated points. Reformulate thi
Analysis III Mat 3120
Tuesday, May 6, 2014
Assignment 1
(due Wednesday, May 14, at the beginning of the lecture)
(1) Let 0 < p < 1. Verity that the
p
-distance
1
p
n
|xi yi |p
dp (x, y) =
i=1
fails the triangle inequality.
(2) An astroid is a planar curve
Analysis III Mat 3120
Solutions to Assignment 1
What are the assignments for?
I have seen many cases where, in this course, a someone with close to a 100 %
assignment record would fail badly and unexpectedly the mid-term and the nal
exam. Hence, a small e
Analysis III Mat 3120
May 14, 2014
Assignment 2
(due Wednesday, May 21, at the beginning of the lecture)
(1) Let X = (X, d) be a metric space, let x X and let r > 0. The sphere of
radius r around x is the following subset of X:
Sr (x) = cfw_y X : d(x, y)
Analysis III Mat 3120
May 5, 2014
Lecture 1. Metric spaces
Recall that the Cartesian square X 2 = X X of a set X is the set formed by all
ordered pairs (x, y), x X, y X.
Denition 1.1. Let X be a non-empty1 set. A metric, or a distance, on X is a realvalue
Analysis III Mat 3120
May 12, 2014
Proposition 3.1. A point x X is a closure point of a subset A of a metric space
(X, d) if and only if for every > 0 the open ball B (x) meets A:
B (x) A = .
Proof. (necessity, or only if part): Suppose x is a closure poi
Analysis III Mat 3120
Solutions to Assignment 2
(1) [4 points] Let X = (X, d) be a metric space, let x X and let r > 0. The
sphere of radius r around x is the following subset of X:
Sr (x) = cfw_y X : d(x, y) = r.
Prove that the sphere Sr (x) is a closed
Analysis III Mat 3120
May 26, 2014
Lecture 6
6.1. Cardinalities. Informally, the cardinality, or the cardinal number, of a set X
is the number of elements in X. It is denoted |X|. Given two sets X and Y , we say
that |X| = |Y | (two sets have the same car
Analysis III Mat 3120
May 21, 2014
Lecture 5
5.1. Everywhere dense subsets.
Theorem 5.1. Let A be a subset of a metric space X. The following conditions are
equivalent.
(1)
(2)
(3)
(4)
cl X A = X.
Every point of X is a closure point for A.
For every x X a
Analysis III Mat 3120
May 14, 2014
Lecture 4
4.1. Some examples.
Example 4.1. Denote J the set of all non-decreasing (that is, non-strictly increasing) bounded real sequences:
J = cfw_x
: m, n N (m n) (xm xn ).
This set is closed in . The easiest strateg
Analysis III Mat 3120
May 7, 2014
Submission date for Assignment 1
It occured to me that making Wednesday the assignment submission day makes
more sense because by Monday you would have your questions ready and can ask
them before or after the Monday lect