Oct 21, 2009.
Taylor Expansion.
A function f (x) that has n derivatives can be written as
1
f (x) = f (a) + f (1) (a)(x a) + f (2) (a)(x a)2 + + f (n) (a)(x a)n + Rn (x)
2
= P n ( x a ) + R n ( x)
How big is Rn (x) ? limxa
Rn (x)
(xa)n
= 0. See this by LH
Assignment 11. Due Dec 2, 2003.
Q1. Consider the space 1 of sequences a = cfw_an : n 1 such that
The distance between two sequences a and b is dened as
n=1
|an | = a < .
|an bn |
d(a, b) =
n=1
Show that a closed subset C
1
is compact if and only if
sup a
Assignment 8. Due Nov 11, 2003
Q 1. For p 1, lp is the space of sequences a = cfw_an with
|an |p < .
1
Check that a p = ( n |an |p ) p is a norm on lp and lp is complete under
this norm.
d
Q 2. If < and < with = d show that > 0 a.e. (as well
as ) and
1
f
Oct 14, 2009.
Power Series.
a n xn = a 0 + a 1 x + a 2 x2 + + a n xn +
f ( x) =
n
It may not converge. It always converges when x = 0. If it converges for some value say
x = x0 , then an xn 0. Therefore |an xn | C . If |x| < |x0 | by comparison
0
0
| a n
If we have two nonnegative functions f (x) and g (x) and we compare their behavior as
x , i.e for large values of x, there are many possibilities. f (x) can get much larger
than g (x) which means
g ( x)
=0
lim
x f (x)
or the other way around g (x) gets mu
1. A monotone class C is one with the the property that if either An A or An A and
An C for every n then A C . A eld F is a class that is closed under nite unions and
complementation. It follows then it is closed under nite intersections as well. A eld is
Assignment 7. Due Nov 4, 2003
Q 1. Consider the following function d(x, y ) dened for irrational real
numbers E R.
d(x, y ) = |x y | +
j =1
1
1
1 | (xrj ) (yrj )|
1
1
2j 1 + | (xrj ) (yrj ) |
where cfw_rj : j = 1, 2, . . . is an enumeration of the rationa
Assignment 3. Due September 30.
1. Given a continuous function f (x) on a closed subset C [0, 1] dene an extension of f
from C to [0, 1] as follows: If x C then there is an interval (a, b) such that a < x < b,
/
(a, b) C = and a, b C . Extend f between a
Assignement 2. Due September 23, 2003
1. Let cfw_xn : n 1 be a sequence of points in [0, 1] and cfw_pn : n 1 a sequence of positive
numbers such that
m=
pn <
n
Show that
(A) =
pn
n:xn A
denes a set function (A) dened for all subsets A [0, 1] which is a c
Assignment 9. Due Nov 18, 2003
Q 1. If (X, B, µ) is a ﬁnite measure space , say for instance µ(X) = 1, then
show that
Lp (X, B, µ) ⊂ Lp (X, B, µ)
for p ≥ p ≥ 1.
Q 2. For a function f ∈ ∩p≥1 Lp (X, B, µ) when is
lim f
p→∞
p
If it is ﬁnite what is its value
Model Questions.
Denite and Indenite Integration
1.
1
x dx
0
1 x
2.
1
x
dx
0
1 x
3.
1
0
x ex dx
4.
x sin x dx
5.
sin2 2x dx
6.
ex sin xdx
7.
dx
0 100+x2
8. Why is
2
0
f (sin x)dx =
9. Can you use it to show
explicit calculation.
10.
1 sinx
e
0
2
0
f (cos
Assignment 5. Due Oct 14, 2003
1. Let f (x) 0 be a continuous function on [0, 1]. We saw that if we dene f (x) by
x+h
1
|h|1 2h
f (x) = sup
f (y )dy
xh
then for some constant C , independent of f
[x : f (x) ]
1
C
f (x)dx
0
Show that a uniform estimate of
Assignment 5. Due Oct 21, 2003
Q 1. We say that F is absolutely continuous in [0, 1] if for any given
> 0 such that
> 0, there exists a
N
|F (bj ) F (aj )|
j =1
when ever the intervals [aj , bj ] [0, 1] are disjoint and
N
|bj aj | < .
j =1
Show that if
Assignment 4. Due Oct 7, 2003.
Construct the Lebesgue measure (area) on the Borel subsets of the square cfw_(x, y ) : 0
x 1, 0 y 1 along the foloowing steps.
1. Start with Borel rectangles, i.e. sets of the form cfw_E = (x, y ) : x A, y B where A
and B