Mathematics 121 Midterm Terence Tao April 30, 1997 All questions are of equal value. There is plenty of working space, and a blank page at the end. Good luck!
Full name: Student ID: Signature:
Problem 1. Problem 2. Problem 3. Problem 4. Problem 5. Total:
Mathematics 121 Final
Terence Tao
June 10, 1997
Full name:
Student ID:
Signature:
Problem 1.
Problem 2.
Problem 3.
Problem 4.
Problem 5.
Problem 6.
Problem 7.
Problem 8.
Total
Problem 1. A set X Rn is said to be star-shaped at the origin if for every x X,
CLASS NOTES FOR APRIL 14, 2000
Announcement: Section 1.2, Questions 3,5 have been deferred from Assignment 1
to Assignment 2. Section 1.4, Question 5 has been dropped entirely.
1. Review of Wednesday class
Let (X, d) be a metric space. A set E X is said t
Problem 1. Let l be the space of all bounded sequences of real numbers (xn ) , with
n=1
the sup norm
x = sup |xn |.
n=1
Show that (l , ) is a Banach space. (You may assume that this space satises the
conditions for a normed vector space).
Solution. Since
CLASS NOTES FOR WEEK 8 (MAY 22-26, 2000)
1. Introduction
This week we will cover the topic of product spaces. Recall that the Cartesian
product of two sets X Y is dened as the space of all pairs of elements (x, y) such
that x X, y Y :
X Y := cfw_(x, y) :
Problem 1. Let (X, dX ) and (Y, dY ) be metric spaces. Suppose that there is a bijection
f : X Y such that
1
dX (x1 , x2 ) dY (f (x1 ), f (x2 ) 10dX (x1 , x2 )
10
for all x1 , x2 X.
Show that if X is complete, then Y must also be complete.
[A function f :
Math 121 - Test 1
Sep 29 2011
Question 1:
1
lX2
($03) = scam):
(a)[3 points] Let f(x) =
l
l-(W)L
_L_
lx
we
and g(x) = Determine, simplify, and find the domain of (f o g)(X) .
(b)[3 points] Let H(x) = «3/2 + lxl and g(x) = 2+x. Find functions f and
Math 121 Review for Test 2
1.(Section 3.1) Functions their domain and range; finding the value of a function; calculating sums, differences, products, and quotients of two functions; expressing one variable as a function of another. Chapter 3 Review exerc