Math 421, Homework #4 Solutions
Note: In problems (1) and (2) we will denote the operator norm by L to distinguish it from the usual
norm on Rn .
(1) (8.2.11) Let T L(Rn , Rm ), and dene
M1 := sup
T(x) = sup
x Rn ; x = 1
T(x)
x =1
T(x) C x for all x Rn .
Math 421, Homework #5 Solutions
(1) (8.3.6) Suppose that E Rn and C is a subset of E .
(a) Prove that if E is closed, then C is relatively closed in E if and only if C is a closed set (as
dened in Denition 8.20(ii).
Proof. First assume that C is a closed
Math 421, Homework #3 Solutions
(1) Consider a function f : R R, and assume that f is continuous at x0 R and locally integrable on
R. Prove that
x0 +
1
f (x) dx = f (x0 ).
lim+
0 2 x0
Proof 1. In order to prove that
lim+
0
x0 +
1
2
f (x) dx = f (x0 ).
Math 421, Homework #2 Solutions
(1) Let f : [a, b] R be a bounded function. Assume that f has a nite number of discontinuities, i.e.
assume there exists a nite subset E of [a, b] so that f is continuous at all x [a, b] \ E . Prove that
f is integrable on
Math 421, Homework #1 Solutions
(1) Let f , g : [a, b] R be bounded functions.
(a) Show that
b
b
(f (x) + g (x) dx (U )
(U )
b
f (x) dx + (U )
a
g (x) dx.
a
a
(This is part of problem 5.1.7(a) in the textbook.)
Proof. We start by proving a general fact ab
Math 421, Homework #6 Solutions
(1) Let E Rn Show that
o
c
E = (E c ) ,
i.e. the complement of the closure is the interior of the complement.1
Proof. Before giving the proof we recall characterizations of the interior and closure (proved in
lecture) that
Math 421, Homework #7 Solutions
(1) Let cfw_xk and cfw_yk be convergent sequences in Rn , and assume that limk xk = L and that
limk yk = M. Prove directly from denition 9.1 (i.e. dont use Theorem 9.2) that:
(a) limk xk + yk = L + M.
Proof. Let > 0. The
Math 421
Test I Solutions
February 17, 2010
1. Determine whether each of the following statements is true or false. If a given statement is
true, write the word TRUE (no explanation or proof is necessary). If a given statement is
false, write the word FAL
Math 421, Homework #10 Solutions
(1) (11.1.4) Assume that f : [a, b] [c, d] R is continuous and that g : [a, b] R is integrable. Prove
that
b
F (y ) =
g (x)f (x, y ) dx
a
is uniformly continuous on [c, d].
Proof. Let > 0.
Since g is assumed to be integrab
Math 421, Homework #9 Solutions
(1) (a) A set E Rn is said to be path connected if for any pair of points x E and y E there
exists a continuous function : [0, 1] Rn satisfying (0) = x, (1) = y, and (t) E for all
t [0, 1]. Let E Rn and assume that E is pat
Math 421, Homework #8 Solutions
(1) Find an example of a function f : R2 \ cfw_0 R for which lim(x,y)0 f (x, y ) exists, but the iterated
limits limx0 limy0 f (x, y ) and limy0 limx0 f (x, y ) do not exist.
Answer. Note that many correct answers are possi
M421 HW 4 Due Friday Oct. 26
From Wade Section 9.2 9.3 9.4 Page Number 269-270 275-277 279 Problems 3(ab) 4, 5, 8 2, 3, 6
Non-book Exercises
1) For two sets A, B Rn define dist(A, B) =
xA,yB
inf
x-y .
(a) Show that if E Rn is closed and K Rn i
M421 HW 3 Due Friday Oct. 5
From Wade Section 8.3 8.4 9.1 Page Number 248 254 262 Problems 3, 5 9ab, 10a 4, 7, 8
Non-book Exercises
1) Suppose that A B Rn . Prove that A B and A B . For the following exercises, the lp norm is defined on Rn by
n
M421 Analysis II
Syllabus
Professor: Keith Promislow, D212 Wells Hall, 432-7135, kpromisl@math.msu.edu Office Hours: Th 10:00-11:00am, W 4:00-5:00pm, or by appt. Text: An Introduction to Analysis, 3rd ed., William Wade CH 5 CH 8 CH 9 Integrability o
M421 HW 2 Due Friday Sept. 21
From Wade Section 5.2 5.3 Page Number 124-125 131-133 Problems 2, 3, 5a, 8 3c, 4abd, 8
Non-book Exercises
1) For n = 1, 2, . . . , define gn (x) = 2xne-nx for x [0, 1]. (a) Show that x [0, 1], lim gn (x) = 0.
n
2
(b)
M421 HW 1 Due Friday Sept. 7
From Wade Section 5.1 Page Number 114-115 Problems 2b (note Pn given in 2a), 3, 4, 5, 6
Non-book Exercises
1) Complete the proof of Remark 5.7. Show that if f : [a, b] R is bounded, and P, Q P[a, b] satisfy Q P then U
M421 Exam 2 Monday Nov. 13
NAME:
1
2
3
4 TOTAL
The value of each question is indicated next to the question. No calculators, books, or notes are permitted. Except where indicated otherwise, you may use any Theorem we have proved in this class,
£062}? M421 HW 1 i
Due Friday Sept. 8
From Wade
Section Page Number Problems
5.1 114115 2b (note Pn given in 2a), 3, 4, 5, 6
Non-book Exercises
1) Complete the proof of Remark 5.7. Show that if f : [(1,1)] H R is bounded, and
HQ 6 P[a,b] satisfy Q Q
Math 421
Test II Solutions
March 24, 2010
1. Determine whether each of the following statements is true or false. If a given statement is
true, write the word TRUE (no explanation or proof is necessary). If a given statement is
false, write the word FALSE