Math 401
Final Exam Review
The following problems are representative of the problems that will appear on the
Final Exam on Tuesday the 12th. Justify your answers.
For problems 5 and 6, if you construct a continuous function, you need not prove
that it is
Math 401
Homework #15 due 11/12
Reading Assignment:
Read sections 3.2 & 3.3 of Understanding Analysis.
Go over your class notes and make sure that you know and understand the following
denitions:
1. A limit point x of a subset A of R.
2. An isolated point
Math 401
Homework #16 due 11/14
Reading Assignment:
Read sections 3.3 of Understanding Analysis.
Go over your class notes and make sure that you know and understand the following
denitions:
1. A compact subset of R.
2. An open cover of a subset A of R.
3.
Math 401
Homework #18 due 11/19
Reading Assignment:
Read sections 3.5 of Understanding Analysis. We will only cover the parts of this
section that concern dense and nowhere-dense sets. We will not cover G-sigma sets
or Baires Theorem.
Go over your class n
Math 401
Homework #19 due 10/01
Reading Assignment:
Read sections 4.1-4.3 (pages 99-113) of Understanding Analysis and make sure the
thou understand the various denitions of the following:
1. For a function f : A R and a limit point c of the domain A, the
Math 401
Homework #21 due 12/3
Reading Assignment:
Read section 4.4 (pages 114-119) of Understanding Analysis and make sure the thou
understand the following theorems and denitions:
1. Denition 4.4.1. Given a function f : A R and B A, then f (B ) denotes
Math 401
Homework #1 due 9/26
Reading Assignment:
1. Read Chapter 1 (pages 3-5 )of Functional Analysis available through the link
in the email.
General Questions (not to be handed in):
1. Why is it important to understand that a continuous function can be
Math 401
Homework #2 due 9/28
Reading Assignment:
Read over your class notes and go over the following two equivalent denitions:
1. A sequence a converges to a limit A, denoted cfw_ai A, if for each real
i=0
number > 0, there exists some term in the sequ
Math 401
Homework #2 due 9/28
Reading Assignment:
Read section 2.2 (pages 38-43) of Understanding Analysis.
Go over your class notes and write out denitions of the following:
1. A function f : R R is continuous at a point x = A.
General Questions (not to
Math 401
Homework #4 due 10/03
Reading Assignment (not to be handed in):
Go over your class notes and write out denitions of the following:
1. The distance from a point a to 0 in Rn , i.e. |a|.
2. The distance between two point a and b in Rn , i.e. |a b|.
Math 401
Homework #6 due 10/10
Reading Assignment:
Read sections 1.1 and 1.2 (pages 1-11) of Understanding Analysis.
Go over your class notes and write out denitions of the following:
1. The set of natural numbers N.
2. The set of integers Z.
3. The set o
Math 401
Homework #5 due 10/05
Reading Assignment:
Go over your class notes and write out denitions of the following:
1. Hilbert space H.
2. The norm | | on H.
General Questions (not to be handed in):
1. Can you think of another interesting aspect of the
Math 401
Homework #7 due 10/12
Reading Assignment:
Read sections 1.1 and 1.2 (pages 13-17) of Understanding Analysis.
Go over your class notes and write out denitions of the following:
1. An upper bound of a subset of real numbers A.
2. The least upper bo