MATH 4010 (2014-15)
Functional Analysis
CUHK
Suggested Solution to Homework 5
Yu Mei
P140, 7. Show that in an inner product space, x y if and only if we have kx + yk = kx yk for all scalars
.
Proof. I
MATH 1010
FINAL EXAM
MAY 10, 2012
NAME: _ A NSW E l2 KE\l
IMPORTANT: When you start, tear out the last two pages, which state full
version of the exam problems, including necessary denitions and lemma
REAL ANALYSIS I BASIC MEASURE AND INTEGRATION
This course will follow a near minimal path through those aspects of the theory
of Lebesgue measure and integral used in virtually all applications in ana
MATH414 FUNCTIONAL ANALYSIS
HOMEWORK 5
Hw 5: 5/140 8,9/141 1,2/150
Problem (5/140)
Show that for a sequence (xn ) in an inner product space the conditions |xn |
|x| and hxn , xi hx, xi imply converge
Measure zero and the characterization of Riemann integrable
functions
Jeffrey Schenker
July 25, 2007
Let us define the length of an interval I (open or closed) with endpoints a < b to be
`(I) = b a.
(
REAL ANALYSIS II HOMEWORK 2
CIHAN
BAHRAN
Stein & Shakarchi, Chapter 4
2. In the case of equality in Cauchy-Schwarz inequality we have the following. If
| hf, gi | = kf k kgk and g 6= 0, then f = cg fo
MATH5011 Suggested Solution to
Exercise 2
(1) Let f be a non-negative measurable function.
(a) Prove Markovs inequality:
Z
n
o
1
x X : f (x) M
f d,
M X
for all M > 0.
(b) Deduce that every integrabl
MATH31011/MATH41011/MATH61011:
FOURIER SERIES AND LEBESGUE INTEGRATION
Chapter 3: Measure and Lebesgue Integration
Revision of Riemann Integration
One of the most important operations in mathematics i
Math 563 - Fall 15 - Homework 3 - selected solutions
4. Prove that a random variable X is independent of itself if and only if there
is a constant c such that P (X = c) = 1. Hint: What can you say abo
Math 563 - Fall 15 - Homework 1
6. (from Resnick) Let P be a probability measure on B(R), the Borel sets in
R. Prove that for any E B(R) and any > 0 there exits a finite union of
disjoint intervals A
Math 563 - Fall 15 - Homework 2 - Solution
4. (from Durrett) Let X be a real valued function on and let (X) be
the -field generated by the sets X 1 (B) where B is a Borel set in R. (This
is the smalle
Math 563 - Fall 15 - Homework 4
5. (Resnick) Let X be a random variable. A number m is called a median of
X if P (X m) 1/2 and P (X m) 1/2. You should convince yourself
that there is always at least o
Math 563 - Fall 15 - Homework 6 Solutions
1. (from Durrett) Let Xn be a sequence of integer valued random variables,
X another integer valued random variable. Prove that Xn converge to X in
distributi
Math 563 - Fall 15 - take home final solutions
1. Let An be a sequence of independent events such that limn P (An )
exists. Prove that
1 X
[1A P (Ak )]
n k=1 k
n
converges in distribution to a normal
HOMEWORK 3 FOR 18.100B/C, FALL 2010
SOLUTIONS
As usual, physical homework due in 2-108 by 11AM. Electronic submission (to
rbm at math dot mit dot edu) up to 5PM.
HW3.1 Let X be a set with the discrete
04.05
Graphing Exponential Functions
Social Sharing
Three Algebra 1 students are comparing how fast their social media posts have spread. Their results are
shown in the following table.
Student
Descri
Some Important Results from Analysis
H. Krieger, Mathematics 156, Harvey Mudd College
Fall, 2008
Denitions: Let cfw_an be a sequence of real numbers. Then we let
lim inf an = lim ( inf a ) = sup( inf
V
V
fi ( j ) =
ij
V
F
V
F
F
B = cfw_ (1 , , n
1 ,i = j
ij =
0 , i 6= j
V
V
fi
i
B = cfw_f1 , , fn
B
B
0 = c1 f1 ( i ) + + cn fn ( i ) = ci
B
V .
B
V
V
B = cfw_f1 , , fn
f = c 1 f1 + + c n fn = 0
f
THE WEIERSTRASS PATHOLOGICAL FUNCTION
Until Weierstrass published his shocking paper in 1872, most of the mathematical world
(including luminaries like Gauss) believed that a continuous function could
POWER SERIES ARE CONTINUOUS
Let (an ) be a sequence in C. Let = lim sup |an |1/n , and assume that < . Then, setting
R = 1/ (or R = + in the case = 0), we know that the series
f (z) =
X
an z n
(1)
n=0
Problem Set #10
Math 471 Real Analysis
Assignment: Chapter 8 #2, 3, 6, 8
Clayton J. Lungstrum
December 1, 2012
Exercise 8.2
Prove the converse of Holders inequality for p = 1 and p = . Show also that
CONTINUOUS ALMOST EVERYWHERE
Definition 1. Let be a subset of R. Say that has
S measureP0 if, for each > 0, there is a
sequence of balls (Bj )
with
radii
r
,
such
that
j
j=1
j Bj and
j rj < .
Given a
1. 20) Construct an example of a sequence of nonnegative measurable
functions from R to R that shows that strict inequality can result in
Fatous Lemma.
Consider the sequence of functions fn (x) = [n,n
CONSTRUCTION OF R
1. M OTIVATION
We are used to thinking of real numbers as successive approximations. For example,
we write
= 3.14159 . . .
to mean that is a real number which, accurate to 5 decimal
CONDITIONAL CONVERGENCE
Here is a concrete example of conditional convergence. Recall that the series
X
(1)n+1
S=
n
n=1
converges (by the alternating series test), but not absolutely. (The fact that i
The University of Sydney
School of Mathematics and Statistics
Solutions to Tutorial 6 (Week 7)
MATH3969: Measure Theory and Fourier Analysis (Advanced)
Semester 2, 2014
Web Page: http:/www.maths.usyd.
.
LGEBRA LINEAR
ISBN 978-85-915683-0-7
ROBERTO DE MARIA NUNES MENDES
Professor do Departamento de Matemtica e Estatstica e do
Programa de Ps-graduao em Engenharia Eltrica da PUCMINAS
Belo Horizonte
Ed
Landon Kavlie
Math 711
Homework #4
26-October-2010
1. Show that if f : R R is differentiable everywhere, then f 0 is Borel
measurable.
Proof. First, note that since f is differentiable for all x R, f
Anlise
Moiss Toledo
11 de abril de 2012
Soluo de exerccios: 8.1 8.7 - Seo 8
Exerccio 1. O cone C = cfw_(x, y, z) R2 ; z 0, x2 + y 2 z 2 = 0 homeomorfo a
R2 .
Demonstrao.
(i) Seja a funo bijetiva
:
C
R