Assignment 1, Math 711
Due January 28, in class
1. Suppose that K is an algebraically closed eld and X K. We say
that X is algebraically independent if for all x X, x is not algebraic
over X \ cfw_x. A maximal algebraically independent subset of K is
call
Assignment 3, Math 711
Due February 28, in class
1. Prove that if M is a countable model in a countable language then M
can be elementarily extended to a countable homogeneous model.
2. Show that if Mn are countable models for n N in a countable language
Assignment 2, Math 711
Due February 11, in class
1. We call a class of L-structures elementary if it is the class of models of
some set of L-sentences. Prove that a class is elementary i it is closed
under isomorphisms, elementary submodels and ultraprodu
EXERCISES ON FUNCTIONS OF BOUNDED VARIATION
AND ABSOLUTELY CONTINUOUS FUNCTIONS
We will write f BV ([a, b]) if f is a function of bounded variation on [a, b],
and f AC([a, b]) if f is absolutely continuous on [a, b].
1. (a) Let f BV ([a, b]). Show that f
Assignment 4, Math 711
Due Apr. 1, in class
1. We shall prove the Craig interpolation theorem. That is, suppose we
have two languages L1 and L2 and L = L1 L2 . Further suppose that
there are sentences in L1 and in L2 such that |= i.e. any
L1 L2 -structure
MATH 711 Real Analysis
Take-home Final, Fall 2014
X will denote a normed vector space over the real numbers; its dual space (the space of
continuous linear functionals from X to IR) will be denoted X .
The distance from a point x X to a subset S X is d(
SINGULARITY AND ABSOLUTE CONTINUITY
1. (a) What is the Lebesgue decomposition of the Dirac measure x with
respect to Lebesgue measure ?
(b) What is the Lebesgue decomposition of with respect to x ?
2. Suppose is counting measure on B(R).
(a) Is < ?
(b) Is
THE HARDY-LITTLEWOOD MAXIMAL FUNCTION
In these exercises, Hf will denote the Hardy-Littlewood maximal function
of a function f L1 :
loc
Hf (x) (Hf )(x) = sup
r>0
1
m(B(r, x)
|f (y)| dy.
(1)
B(r,x)
1. Let f : R R be dened by
0,
f (x) =
x1/2 ,
if x 0 ,
if x