Problems in Real Variables, II (Math608), Solutions
Prof.: Thomas Schlumprecht
Problem 1. Show that every innite dimensional Banach space admits nets
(xi )iI which converge weakly to 0, but are conally unbounded, which
means that for any i I and any C > 0
Real Analysis Qualifying Exam; January, 2009.
Work as many of these ten problems as you can in four hours. Start each problem on a new
sheet of paper.
#1. Let F Rn be compact and prove that the convex hull conv(F ) is compact. (You may
use without proof t
Real Analysis Qualifying Exam, May, 2008
1. Prove that if E is a closed linear subspace of L2 (0, 1) and each element in E is bounded
(i.e. f L (0, 1) for all f E), then E is nite dimensional.
2. Let fn :=
2n
k
k=1 (1) ( k1 , 2k ] .
n
2n
Prove that fn 0 w
Real Analysis Qualifying Exam, May, 2007
Directions: Work each problem on a separate piece of paper.
1. Let (X, , ) be a nite measure space and let cfw_fn be a sequence of measurn=1
able functions on X. Prove that if cfw_fn converges to zero -almost eve
Problems in Real Variables, II (Math608), Solutions
Prof.: Thomas Schlumprecht
Problem 1. A linear functional on a normed linear space is bounded if and
only if f 1 (cfw_0) is closed.
Proof. is clear since bounded linear functionals on a normed linear spa
Problems in Real Variables, II (Math608), Solutions
Prof.: Thomas Schlumprecht
Problem 1. Dene
c0 = cfw_(xi )iN F : lim xi = 0 (with F = R or C).
i
For x = (xi )iN c0 dene x = supiN |xi |. Show that is norm on co
and that c0 is a Banach space.
Proof. Usin
Problems in Real Variables, II (Math608), Solutions
Prof.: Thomas Schlumprecht
Problem 1. Assume that X is a n.l.s. and that Y is a closed proper
subspace of X. For x X dene
|x + Y | = inf |x + y|.
yY
a) Show that the map | | is a norm on X/Y .
We recall
Problems in Real Variables, II (Math608), Solutions
Prof.: Thomas Schlumprecht
Problem 1. Determine whether or not the following functions are of
bounded variation on [1, 1].
(a)
f (x) = x2 sin(1/x2 ),
(b)
f (x) = x2 sin(1/x),
x = 0, f (0) = 0,
x = 0, f (
Problems in Real Variables, II (Math608), Solutions
Prof.: Thomas Schlumprecht
Problem 1. Problem 11/page 92.
Let be a positive measure on the measurable space (X, M). A collection
of functions (f )A is called uniformly integrable if for every > 0 there i
Problems in Real Variables, II (Math608), Solutions
Prof.: Thomas Schlumprecht
Problem 1.
Assume that X and Y are Banach spaces and that T
L(X, Y ). Show that T is an isomorphism/isometry from X onto Y if and
only if T is an isomorphism/isometry from X o
Problems in Real Variables, II (Math608), Solutions
Prof.: Thomas Schlumprecht
Problem 1. There is a meager subset of R whose complement has Lebesgue
measure zero.
Proof. Given any > 0 we will construct a fat cantor set S [0, 1],
which is closed, nowhere
Problems in Real Variables, II (Math608), Solutions
Prof.: Thomas Schlumprecht
Problem 1.Let X be a non empty set. We call a set F P(X) \ cfw_ a
lter on X if
a) for all A, B F there is a C F so that C A B, and
b) if A F and B A, then also B F.
If only (a)
Real Analysis Qualifying Exam; August, 2009.
Work as many of these ten problems as you can in four hours. Start each problem on a new
sheet of paper.
#1. Evaluate the iterated integral
x exp(x2 (1 + y 2) dx dy.
0
0
(Justify your answer.)
#2. Let f C[0, 1]