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 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
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 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 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 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)