5.5 - Hilbert Spaces
Morgan Weiss
5.19 The Schwarz Inequality - |hx, yi kxkkyk for all x, y H with equality if and only if x and y are
linearly dependent.
Proof. If hx, yi = 0, the result follows. Suppose hx, yi =
6 0 (in particular y 6= 0), let = sgnhx,
3.4 - Differentiation on Euclidean Space
Morgan Weiss
Lemma 3.15 - Let C be a collection of open balls in Rn , and let U =
Pk
disjoint B1 , . . . , Bk C such that 1 m(Bj ) > 3n c.
S
BC
B. If c < m(U ), there exist
Proof. If c < m(U ) by Theorem 2.40 there
5.3 - The Baire Category Theorem and Its Consequences
Morgan Weiss
5.9 The Baire Category Theorem - Let X be a completeTmetric space.
a.) If cfw_Un
1 is a sequence of open dense subsets of X, then
1 Un is dense in X.
b.) X is not a countable union of now
3.1 - Signed Measures
Morgan Weiss
Proposition
3.1 - Let be a signed measure on (X, M ). If cfw_Ej is an increasing sequence inT M , then
S
( 1 Ej ) = limj (Ej ). If cfw_Ej is a decreasing sequence in M and (E1 ) < then ( 1 Ej ) =
limj (Ej ).
Proof. Le
5.2 - Linear Functionals
Morgan Weiss
Proposition 5.5 - Let X be a vector space over C. If f is a complex linear functional on X and u = Ref ,
then u is a real linear functional, and f (x) = u(x) iu(ix) for all x X . Conversely, if u is a real linear
func
3.3 - Complex Measures
Morgan Weiss
The Lebesgue Radon Nikodym Theorem - If is a complex measure and is a -finite measure and
is a -finite positive measure on (X, M ), there exist a complex measure and f L1 () such that
and d = d + f d. If also 0 and d
3.2 - The Lebesgue-Radon-Nikodym Theorem
Morgan Weiss
The Lebesgue Radon Nikodym Theorem - Let be a -finite measure and a -finite positive measure
on (X, M ). There exists a unique -finite signed measure , on (X, M ) such that
and = +
Moreover, there is
5.1 - Normed Vector Spaces
Morgan Weiss
Theorem 5.1 - A normed vector space X is complete if and only if every absolutely convergent series in X
converges.
P
P
Proof. Suppose X is complete and 1 kxn k < . Let > 0. Then choose an N such that n=N kxn k <
.
MAA5616 Chapter 2 Proofs
Morgan Weiss
Background Information:
Any mapping f : X Y between two sets induces a mapping f 1 : P (Y ) P (X), defined by f 1 (E) =
cfw_x X : f (x) E, which preserves unions, intersections, and complements.
Thus, if N is a -algeb
3.5 - Functions of Bounded Variation
Morgan Weiss
Theorem 3.23 - Let F : R R be increasing, and let G(x) = F (x+ ).
a.) The set of points at which F is discontinuous is countable.
b.) F and G are differentiable a.e., and F 0 = G0 .
Proof. a.) Stackexchang
1.3 #11 Let A 2X be an algebra, A the collection of countable
unions of sets in A, and A the collection of countable intersections
of sets in A . Let 0 be a premeasure on A and the induced outer
measure.
(a) For any E X and > 0 there exists A A with E A a
2.4 #44 (Lusins Theorem) If f : [a, b] C is Lebesgue measurable and > 0, there is a compact set E [a, b] such that (E c ) <
and f E is continuous (Hint: Use Egoros theorem and Theorem 2.26)
Proof. First, note that
cfw_|f | n
[a, b] =
nN
and so, by continu
2.5 #49 (The Fubini-Tonelli theorem for Complete Measures)
Let (X, M, ) and (Y, N , ) be complete, -nite measure spaces, and
let (X Y, L, ) be the completion of (X Y, M N , ). If f
is L-measurable and either (a) f 0 or (b) f L1 (), then fx is N measurable
1.5 #29 Let E be a Lebesgue measurable set.
(a) If E N where N is the non measurable set described in section
1.1, then m(E ) = 0
(b) If m(E ) > 0, then E contains a non measurable set. (It suces to
assume E [0, 1]. In the notation of section 1.1, E = rR
1.3 #11
A nitely additive measure is a measure if and only if
it is continuous from below as in Theorem 1.8c. If (X ) < , then
is a measure if and only if it is continuous from above as in Theorem
1.8d.
Proof. Let be a nitely additive measure dened on a
MAA 5616 Final Exam
Name:
Complete two of the following problems (circle the numbers
of the two you want graded):
Problem 1
Suppose f is a nonnegative measurable function on a
measure space (X, M, ) satisfying f d < . Show that for every
> 0 there exists
1.2 #5 If M(E ) is the -algebra generated by E , then M(E ) is the
union of the -algebras generated by F as F ranges over all countable
subsets of E . (Hint: Show that the latter object is a -algebra).
Proof. Let
M=
M(F )
F E
F is countable
so that our ta
2.1 #10
The following implications are valid if and only if the
measure is complete
(a) If f is measurable and f = g -a.e. then g is measurable
(b) If fn is measurable for n N and fn f -a.e. then f is measurable
Proof.
if:
(a) We need to show that for eve
2.2 #15
, then
If cfw_fn L+ , fn decreases pointwise to f , and
n=1
(1)
f d = lim
n
f1 d <
f d.
Proof. Note that cfw_f1 fn L+ is increasing and so
n=1
f1
f=
f1 f =
= lim
n
f1
lim f1 fn = lim
n
fn =
n
f1 lim
n
f1 fn
fn
where the third identity follows
MAA 4224Complex Analysis, Fall 2012
Practice Test
These are practice problems for the midterm. There will be four problems
of this kind on the test.
A. Find the Re(z ), Im(z ), |z |, 1/z , z for the following.
1. 25i/(1 + i)
2. 3e2i/3
3. (1 3i)50
4.
5i
2+
MAA 4402Complex Analysis, Fall 2012
Practice Final
These are practice problems for the nal. You will also be responsible for
the topics covered in the quizzes and midterms. The nal is cummulative.
A. Describe at least three methods from class to determine