Syllabus
MAA 4402
COMPLEX VARIABLES
Fall 2014
SYLLABUS CHANGE POLICY. Except for changes that substantially affect implementation
of the evaluation (grading) statement, this syllabus is a guide for the course and is subject
to change with advanced notice.
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