Real Analysis - Math 630
Homework Set #9 - Chapters 5 and 6
by Bobby Rohde 11-10-00
Problem 12
Let f be absolutely continuous (AC) in the interval , 1 , > 0. Does the continuity of f at 0 imply that f is AC on 0, 1 ? What if f is also a bounded variation
Real Analysis - Math 630
Homework Set #8 - Chapter 5
by Bobby Rohde 11-03-00
Problem 7
a) Let f be of bounded variation (BV) on a, b . Show that for each c a, b the limit of f x exists as x c and also as x c . Prove that a monotone function (and hence a f
Royden, Real Analysis 3rd ed. Chapter 2
Problem 2-11 a. If |xn l| for n N, we get immediately |xn xm | |xn l| + |xm l| 2 for n, m N. b. Equation (1) implies |xn | |xN | + 2 for all n N. Hence cfw_xn is bounded by max |xi | + 2. 1iN c. Assume lim xnm = l.
MTH536 Chapter 8 Problem 4
Gene Quinn
p. 1/
Chapter 8 Problem 4
Prove that a mapping is continuous if and only if the inverse image of every closed set is closed.
p. 2/
Proof of Proposition
Let be an arbitrary closed subset of hypothesis, is continuou
MTH536 Chapter 8 Problem 3
Gene Quinn
p. 1/
Chapter 8 Problem 4
Prove that a set set such that is open if and only if, given
, there is an open
p. 2/
Proof of Proposition
Suppose every is contained in some open set
. Then
p. 3/
Proof of Proposition
MTH536 Chapter 8
Gene Quinn
p. 1/
Chapter 8
If
is a collection of topologies for a set
, then
is a topology for
.
p. 2/
Proof of Proposition
For criteria:
to be a topology for
, it must satisfy the following three
$
!
" #
%
" #
"&
!
p.
MTH536 Chapter 7 Problem 28b
Gene Quinn
p. 1/
Chapter 7 Problem 28b
If space
is a uniformly continuous map of a totally bounded metric onto a metric space , then is totally bounded.
Is this proposition true if continuous is substituted for uniforml
MTH536 Chapter 7 Problem 28a
Gene Quinn
p. 1/
Chapter 7 Problem 28a
Show that if is a uniformly continuous map of a totally bounded metric space onto a metric space , then is totally bounded.
p. 2/
Proof of proposition
In problem 23b, we established t
MTH536 Chapter 7 Problem 23d
Gene Quinn
p. 1/
Chapter 7 Problem 23d
Show that every totally bounded metric space is separable.
p. 2/
Proof that a totally bounded metric space is separable
Suppose is a totally bounded metric space. We must show that has
Royden, Real Analysis 3rd ed. Chapter 3
Problem 3-4 check Assume E1 , E2 , . . . are mutually disjointed, it is easy to n n [ X |Ei | (1) Ei =
i=1 i=1
It is easy to see that (2) holds when the RHS is . Otherwise, all Ei = for i larger than some integer,
Royden, Real Analysis 3rd ed. Chapter 4
Problem 4-2 a. Let h be the upper envelop of f . It is known from Problem 2-51 that h(y ) = max f (y ), xy f (x) . lim If f is a step function, it is clear from (1) that (y ) lim f (x) h(y )
xy
(1)
(2)
at each conti
Royden, Real Analysis 3rd ed. Chapter 5
Problem 5-3 a. Obvious. But it should be D+ f (c) D+ f (c) 0 D f (c) D f (c) . b. At the end point a we can only have D+ f (a) and D+ f (a) . If f has a local maximum at a, we have D+ f (a) D+ f (a) 0.
Problem 5-4 W
Royden, Real Analysis 3rd ed. Chapter 6
Problem 6-1 For any > 0, we have from the denition of kk that |f (x)| kf k + and |g (x)| kg k + . for x a.e. Hence |(f + g) (x)| kf k + kg k + 2 a.e. and thus kf + gk kf k + kgk + 2. Now let 0. Problem 6-2 Let E = c
Royden, Real Analysis 3rd ed. Chapter 7
Problem 7-3 b. Let Xx = fy 2 X : (y; x) < 1g : Then (i) Xx is open. For each y 2 Xx; we have (y; x) < 1: By using the triangle inequality it can be shown easily that B (y; r ) Xx for any r > 0: c c c (ii) Xx is open
Math 515 Professor Lieberman September 7, 2004 HOMEWORK #2 SOLUTIONS Chapter 3 4. To show that n is countably additive, let Ej be a sequence of disjoint sets. We consider three possibilities. (A complete proof requires a lot more set theory than I want to
ROYDEN, REAL ANALYSIS 3RD ED. CHAPTER 8
Problem 8-10 a. Let us assume A1 \ A2 = . For x 2 A1 [ A2 ; x 2 A1 or x 2 A2 . Enough to consider the case x 2 A1 . Then x 2 A2 . Or x 2 A n A2 which is open in A. = Hence there exists an open set O, such that (1) x
ROYDEN, REAL ANALYSIS 3RD ED. CHAPTER 9
c c Problem 9-2. cfw_Kn : n 1 and 0 form an open covering of K1 . Suppose N Kni O 1 c K1 . Then KnN O K1 KnN . Hence O KnN .
Problem 9-4. Let A, B be closed subsets. Note that A, B are compact. Fix x A. By Problem 9
Homework 6: solutions
1. Problem 2, page 119. We have |f (x)| f
except possibly on a set of measure zero. Then
p p 1
f thus f
p
=
0
|f |p
0
1
f
p
=f
p ,
f
.
On the other hand,
f
= inf cfw_M : mcfw_x [0, 1] : |f (x)| > M = 0, > 0 such that f
and therefo
MTH536 Chapter 7 Problem 23c
Gene Quinn
p. 1/
Chapter 7 Problem 23c
Show that total boundedness is not a topological property.
p. 2/
Proof of proposition
We can establish the proposition by exhibiting a homeomorphism from a totally bounded metric space
MTH536 Chapter 7 Problem 23b
Gene Quinn
p. 1/
Chapter 7 Problem 23b
Show that total boundedness is a uniform property.
p. 2/
Proof of proposition
We need to show that total boundedness is preserved by a uniform homeomorphism.
Let let
and
be metric s
MTH536 Chapter 7 Problem 23a
Gene Quinn
p. 1/
Chapter 7 Problem 23a
Show that boundedness is a metric but not a uniform property.
p. 2/
Proof of proposition
The proposition will be established if we can produce a uniform homeomorphism between a bounded
Real Analysis - Math 630
Homework Set #2 - Chapter 3
by Bobby Rohde 9-14-00
Problem 9
Show that if E is a measurable set, then each translate E + y of E is also measurable. Proof
E is measurable so A, we have m*A = m*(A E) + m*(A E). We also know from Pro
Real Analysis - Math 630
Homework Set #3 - Chapter 3
by Bobby Rohde 9-21-00
Problem 18
Show that (v) does not imply (iv) in Propostion 18 by constructing a function f such that cfw_x : f(x) > 0 = E, a given non-measurable set, and such that f assumes each
Real Analysis - Math 630
Homework Set #4 - Chapter 4
by Bobby Rohde 9-28-00
Problem 1
a) Show that if
fx
0, x is irrational 1, x is rational
Then
R
b a
fx
b
x
b
a and R
b a
fx
x
0
Proof
R a f x x inf a x x, step functions (x) f(x), x. But each step functi
Real Analysis - Math 630
Homework Set #5 - Chapter 4
by Bobby Rohde 10-05-00
Problem 3
Let f be a nonnegative measurable function. Show that that f 0 a.e. Proof
BWOC, suppose f 0 a.e. Then
f
0 implies
some set E with mE > 0 such that f x
0,
x
E.
1 0 x fx
Real Analysis - Math 630
Homework Set #6 - Chapter 4
by Bobby Rohde 10-12-00
Problem 10
a) Show that if f is integrable over E then so is f and
Ef E
f
Does the integrability of f imply integrability of f? Proof
f is integrable f and f are each integrable.
Real Analysis - Math 630
Homework Set #7 - Chapter 5
by Bobby Rohde 10-19-00
Problem 1
Let f be the function defined by f 0 0 and f x Find D f 0 , D f 0 , D f 0 , and D f 0 .
D f0
x sin
1 x
for x
0.
D f 0 = lim
h0
fh
can find a value of
D f0
h h0 1 h whic
Real Analysis - Math 630
Homework Set #10 - Chapter 6
by Bobby Rohde 11-17-00
Problem 7
a) For 1 p that v 1 v
v v
p
, we denote l p the space of all sequences v v 1 such . Prove the Minkowski ineqality for sequences
p v p v p.
Here we have 1
v p p
p
v1
,
Real Analysis - Math 630
Homework Set #11 - Chapter 6 & 11
by Bobby Rohde 11-25-00
Problem 6.21
a) Let g be an integrable function on 0, 1 . Show that 0 and measurable function f such that f
f g g
1
a bounded
f
Proof
Case Let
g
1
0 sgn g x , where sgn 0 0
Real Analysis - Math 630
Homework Set #12 - Chapter 12
by Bobby Rohde 11-25-00
Problem 12.2
Assume that Ei is a sequence of disjoint measurable sets and E = Then A we have that
A E A Ei
Ei .
Proof
If Ei has only one non-empty set element then this is triv