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
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 d
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
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
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 a
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
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
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 [c
Julian Febres
Success is no accident. It is hard work,
perseverance, learning, studying,
sacrifice and most of all, love of what you
are doing or learning to do.
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Some people tell me that we pro
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 i
Julian Febres
1/23/2017 Senior Night
Dear Mario,
Its hard to believe how long we have actually known each other. I honestly dont know if
I met you in the math club in middle school or if I already kne
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
2/27/20
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
D
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 (Secti
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 F
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
Domai
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. Th