MATH 605, HW 1 SOLUTIONS
Follands Real Analysis; Chapter 1:
4.) This follows since any countable union can be written as an increasing countable union:
Ej = j =1 Ek ;
j =1
j =1
k
note that j =1 Ek is a nite union of sets in the algebra and is hence in th
MATH 605, HW 2 SOLUTIONS
Follands Real Analysis; Chapter 1:
18.) (a) By denition, given > 0, there exist Aj A (j = 1, 2, . . . ) with E Aj and
1
0 (Aj ) (E ) + . Subadditivity of and Proposition 1.13 give
(Aj )
(Aj ) =
0 (Aj ) .
(b) To get one way, use
MATH 605, HW 6 SOLUTIONS
Follands Real Analysis; Chapter 5:
1.) Let X be a normed v.s. over R.
(a) Show that vector addition and scalar multiplication are continuous from X X and
X R to X : Ill just do this for addition. Dene T : X X X by T (x, y ) = x +
MATH 605, HW 4 SOLUTIONS
Follands Real Analysis; Chapter 2:
19.) Suppose fn L1 () and fn f uniformly.
(a) Show that if (X ) < , then f L1 () and fn f : f is automatically measurable
since fn f . Given > 0, there exists N so that |fn f | < /(X ) for all n
MATH 605, HW 5 SOLUTIONS
Follands Real Analysis; Chapter 2:
39.) Suppose fn f almost uniformly.
(a) Show fn f a.e.: This is immediate. Let B be the bad set where fn does not converge
to f (we say in an earlier assignment that this was measurable). It foll
MATH 605, HW 7 SOLUTIONS
Follands Real Analysis; Chapter 6:
7.) If f Lp L , then show: |f | = limq |f |q . We can assume |f |p = 0 and
|f | = 0 (otherwise f = 0 a.e. so its obvious).
(a) Proposition 6.10 gives |f |q |f |p/q |f |1p/q . Since limr0 ar = 1 a
MATH 605, HW 3 SOLUTIONS
Follands Real Analysis; Chapter 2:
2.) f, g : X R are measurable.
(a) Show that f g is measurable: Since
cfw_x | f g (x) > a = bQ+ [cfw_x | f (x) > b cfw_x | b g (x) > a]bQ [cfw_x | f (x) < b cfw_x | b g (x) > a] ,
we can write (f
MATH 605, HW 8 SOLUTIONS
Follands Real Analysis; Chapter 3:
22.) Suppose f L1 (Rn ) and |f |L1 = 0. Show there exist C, R > 0 so (the maximal function) Hf (x) C |x|n for |x| > R: Since |f |L1 = 0, we can choose R > 0 so
BR/2 (0) |f | > > 0 for some . For
Julian Febres
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2.05b Writing about Polynomials
Beginning with the Fundamental Theorem of Algebra, it states that a polynomial
function with a degree of n has a maximum n complex zeros, which includes both
real and imaginary, and has to have at least one complex zero if
Julian Febres
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Module 7 Study Plan
Section
07.01
07.02
07.03
07.04
07.05
Topic
To Do
Anticipated Completion
Date
Arithmetic Sequences
Lesson (Section
9.4 of your text),
Practice Problem,
Submitted
Assignment
2/27/2017
Geometric Sequences
Lesson (Section
9.4 of your text
Section
Topic
To Do
Anticipated
Completion
Date
02.01
Quadratic Functions
Lesson (Section 2.1 of
your text),
Practice Problems,
Submitted Assignment
November 1st,
2016
Polynomial Functions of Higher
Degree
Lesson (Section 2.3 of
your text),
Practice Probl
Module 5 Study Plan
Section
Topic
To Do
Anticipated Completion
Date
05.01
Using Fundamental
Identities
Lesson (Section 5.1
of your text),
Practice Problems,
Submitted
Assignment
12/24/16
Lesson (Section 5.1
of your text),
Practice Problems,
Submitted
Assi
Module 4 Study Plan
Section
Topic
To Do
Anticipated
Completion
Date
04.01
Angles and Their Measures
Lesson (Section 4.1 of
your text),
Practice Problem,
Submitted Assignment
11/23/2016
Trigonometric Functions of Acute
Angles
Lesson (Section 4.2 of
your te
Choice #1: Describe each of the following properties of the graph of the cosine
function, f(theta) = cos(theta), and relate the property to the unit circle definition
of cosine.
Amplitude
Period
Domain
Range
x-intercepts
The amplitude of the graph of the
06.02b Applying the Laws of Sines and Cosines
Law of Sines
There are four situations in oblique triangles that determine whether the law of sines or
cosines be used to solve for parts of triangles. The first two cases call for law of sines and the
second