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

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 <
.

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

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

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

3.1 - 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

5.3 - 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

3.4 - 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

5.5 - 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,