The Lebesgue Covering Lemma
copied from http:/mathblather.blogspot.com/
February 3, 2012
Theorem(Lebesgue Covering Lemma): Given that X is a compact
metric space, let U be a open covering of X . Then there exist > 0 such
that for all x X , the ball B (x)
HW 8
Yu Sun 9074473373
November 9, 2015
1
Chapter 8
Problem. 12
Proof.
(a)
By f (x) is a real function, and symmetric, then we can use the real form of Fourier
Series. By the symmetry, we know that bn = 0, then
a0 =
2
f (x) cos nxdx =
1
an =
cos nxdx =
2
HW 8
Yu Sun 9074473373
November 23, 2015
1
Chapter 8
Problem. 30
Proof.
By Stirling formula
( x1 )x1 2(x 1)
(x + c)
(x + c)
e
= lim x+c1
F (x)
x xc (x)
x (
(x)
)x+c1 2(x + c 1)
e
lim
F (x) =
1 ( x+c1 )x+c1
e
xc ( x1 )x1
e
x+c1
x1
We just need to prove lim
HW 10
Yu Sun 9074473373
December 6, 2015
1
Chapter 9
Problem. 1 If X, Y are normed vector spaces, show that L(X, Y ), then space of all
bounded linear mappings from X to Y with the operator norm, is also normed vector
space.
Proof.
We rst prove it is a ve
HW 11
Yu Sun 9074473373
December 14, 2015
1
Chapter 9
Problem. 9 If f is a dierentiable mapping of a connected open set E RN into RM , and f (x) = 0 for every
x E, prove that f is constant in E.
Proof.
By E is a open set, then for each x E, exists a neigh
MATH 522 Spring 2015
Prof. Xianghong Gong
Oce: 413 Van Vleck, Phone: 262-3545, E-mail: gong@math.wisc.edu
Oce Hours: TRF 2:303:30 PM, or by appointment.
Textbook: Principles of Mathematical Analysis, third edition, by W. Rudin,
McGraw-Hill
Homework: Assig
MATH 521 REVIEW
1. Review on Riemann integrals
We want to study whether a function on an interval [a, b] can be approximated by better
functions on [a, b] such as C functions (i.e. functions that have derivatives of any order),
polynomials, or trig functi
MATH 522 HOMEWORK 1
1. Let f be a bounded function on [0, 1]. Show that f is Riemann
integrable on [0, 1] if and only if it is Riemann integrable on
[1/n, 1] for n = 1, 2, . . . and
1
f (x) dx
lim
n
1/n
exists.
2. Let B1 be the closed unit ball in B(E) de
MATH 522 HOMEWORK 3
1. Determine all values of x such that
1 cos(2kx)
k
k=1
converges at x.
2. Show that the series
k=1
1 cos(x/ k)
k
converges to f pointwise on R. Does the series converge uniformly on R? Is f continuous on R? Does f exist on R?
3. Let f
MATH 522 HOMEWORK 2
1. (i) Show that the sequence sin(nx) does not converge uniformly on [0, 1]. Give two proofs.
(ii) Determine if the series
x
1 + n2 x2
converge uniformly on [0, 1].
2. Let fn , gn be real-valued functions dened on (0, 1). Suppose
that
Math 522, Exam I, March 2, 2015 (90 minutes)
1. Determine if the series
n=1
x
1 + n 2 x2
converges uniformly on [1, ). Determine if it converges uniformly on [0, 1].
Proof. For x 1, we have
x
x
1
2 2 2.
0
2 x2
1+n
nx
n
1
Since
is convergent, by Weierstra
Math 522, Review Problems, May 11 (Final on May 15, 2015)
1. Determine the uniform convergence of
n=1
n=1
(1)n
,
4
n+x
x [1, );
(1)n
,
n + x + (1)n
n=1
(1)n
,
n + x + (1)n
x [1, );
x [1, ).
Hint for the second part: Consider
2N
n=1
(1)n
,
n + x + (1)n
x [
THE BAIRE CATEGORY THEOREM
Want to show for any open set U_0, U_0
cfw_mathcap E_n = empty for all n <=>
infinite intersection of E_n is dense
1. Baires theorem
1.1. Theorem. Let X be a complete metric space X. If En is a sequence of dense open
subsets of
1. Uniform Convergence
A sequence of functions
,
if, for each
, an integer
A series
, 2, 3, . is said to be uniformly convergent to
can be found such that
for
converges uniformly on
if the sequence
for a set of values of
and all
.
of partial sums defined
THE BAIRE CATEGORY THEOREM
1. Baires theorem
1.1. Theorem. Let X be a complete metric space X. If En is a sequence of dense open
subsets of X, then En is dense in X.
Proof. Let U0 be any open ball in X. We want to show that U0 intersects all En . Let U0 =
HW 5
Yu Sun 9074473373
November 3, 2015
1
Chapter 8
Problem. 4 Prove the following limit relations:
bx 1
= ln(b).
x0 x
lim ln(1+x) = 1.
x
x0
lim (1 + x)1/x = e.
x0
x
lim (1 + n )n = ex .
n
(a) lim
(b)
(c)
(d)
Proof.
(a)
By LHospitals rules:
bx 1
ex ln(b)