Dean Baskin
Qual Problems, Fall 2007
Fall 2007 Qual, Part I: 1,3,5; Part II:1,2,4,5
I.1 Two short problems:
1. Suppose that ( X , A, ) is a measure space, and X = 1 X j with X j X j+1 for
j=
all j. Let j be the indicator (characteristic) function of X j ,
Ph.D. Qualifying Exam problems, Real Analysis
June 2005, part I.
1
(Quickies)
Let (B, | |) be a normed space, and A : B B an invertible linear transformation such
that An c for some constant c > 0 and all n Z. Prove that there is an equivalent norm on
B w
Homework # 3.
The rst two problems need not be turned in:
1
1
0.1. Given p > 1 and q > 1 such that +
= 1 prove the Hlder
o
p
q
inequality:
|f g |
|f |p
1/p
|g |q
1/q
.
You may nd this in every book but try to do that on your own. First show
that for any
Homework # 6.
1. The Poisson kernel P (z ) : D1 D1 R (here D1 = cfw_|z | 1 C is
the unit disk) is dened as
P (z, ) = Re
+z
z
=
1 |z |2
, |z | < 1, | | = 1.
| z |2
If D(w, ) = cfw_z C : |z w| < is an open disk, and : D(w, ) R
is an L1 (D) function, then
Homework # 4.
Lq [0, 1]
1. Let g
and dene a mapping G : Lp [0, 1] R by G(f ) =
Show that G is a bounded linear functional on Lp [0, 1], that is, G
is linear and there exists a constant C > 0 so that |G(f )| C f Lp for all
f Lp . Here 1/p + 1/q = 1.
2. Le
Final Exam, Math 205, Fall 2011
1. Let A be a compact subset of L1 [0, 1]. Show that A satises the following condition: for any
> 0 there exists > 0 such that for any f A and any measurable set B with m(B ) < we have
B |f | < .
2. Let H s (Rn ) be the se
Homework # 2.
1. Let be a non-negative continuous function on Rn such that = 1.
Given t > 0 dene t (x) = tn (x/t). Show that if g C (Rn ) with
compact support then
t (g ) =
Rn
t (x)g (x)dx g (0).
Because of that t is called an approximation of identity.
2
Homework # 1.
The ternary expansion of a real number r [0, 1] is its representation
an
as r =
with an = 0, 1, 2. The sequence cfw_an is called the ternary
3n
n=1
expansion of r. Such an expansion is unique (up to a tail of twos) for a
given r [0, 1]. The
Solution Set
Math 205a - Fall 2011
MIDTERM
Problem 1 Let fn be a sequence of functions in L2 ([0, 1]) such that fn
any > 0 there exists N so that for all n N we have
L2
1 for all n. Show that for
mcfw_x [0, 1] : |fn (x)| n2/3 for all n N 1
Here m is th
Ph.D. Qualifying Exam, Real Analysis
September 2005,
part I
Do all the problems.
1
(Quickies)
a. Let B denote the set of all Borel probability measures on [0, 1]? What are the extreme
points of this set?
b. Suppose that u is a continuous linear functional
Ph.D. Qualifying Examination, Real Analysis
Spring 2006,
part I
Do all the problems.
1
Quickies
a. Let fn Lp ([0, 1]) where 1 < p < . Suppose that |fn |p 1 and moreover that
fn (x) 0 for a.e. x. Prove that fn 0 weakly in Lp .
b. Let v1 , . . . , vN be a n
Dean Baskin
Qual Problems, Spring 2007
Spring 2007 Qual, Part I: 1,3,4; Part II: 4
I.1
1. Suppose that X and Y are Banach spaces, and An , n = 1, 2, 3, . . ., and A are
bounded linear operators from X to Y . Suppose also that An A in the weak
operator top
Dean Baskin
Qual Problems, Fall 2006
Fall 2006 Qual, Part I: 1, 2, 3, 4; Part II: 1
I.1 Let G be an unbounded open set in (0, ). Dene
D = cfw_ x (0, ) : nx G for innitely many n.
Prove that D is dense in (0, ).
Consider the sets
Dk = cfw_ x (0, ) : nx G f
Dean Baskin
Qual Problems, Spring 2006
Spring 2006 Qual, Part I: 1, 2, 3, 5; Part II: 1, 2
I.1
1. Let f n L p ([0, 1]), where 1 < p < . Suppose that f n p 1 and moreover that
f n ( x ) 0 for a.e. x [0, 1]. Prove that f n 0 weakly in L p .
2. Let v1 , . .
Dean Baskin
Qual Problems, Fall 2005
Fall 2005 Qual, Part I: 1b, 3, 4; Part II: 1, 4, 5
I.1b Suppose that u is a continuous linear functional on C (T) (i.e. a distribution on
the circle), which has the property that u, 0 whenever 0, C (T). Show
that u is
Dean Baskin
Qual Problems, Spring 2005
Spring 2005 Qual, Part I: 2, 3; Part II: 6, 7, 10
I.2 Prove that a linear operator T on a Hilbert space H is compact if, and only if, it is
the limit in the operator norm topology of a sequence of operators of nite r
Ph.D. Qualifying Exam, Real Analysis
Fall 2007,
part I
Do all the problems. Write your solution for each problem in a separate blue book.
1
Two short problems:
a. Suppose that (X, A, ) is a measure space, and X = Xj with Xj Xj +1 for all
j =1
j . Let j be
Ph.D. Qualifying Exam, Real Analysis
Spring 2007,
part I
Do all the problems. Write your solution for each problem in a separate blue book.
1
Two short problems:
a. Suppose that X and Y are Banach spaces, and An , n = 1, 2, 3, . . ., and A are bounded
lin
Ph.D. Qualifying Exam, Real Analysis
September 2006,
part I
Do all the problems.
1
Let G be an unbounded open set in (0, ). Dene
D = cfw_x (0, ) : nx G
for innitely many n.
Prove that D is dense in (0, ).
2
Suppose that f L1 ([0, 1]) but f L2 ([0, 1]). Fi
Homework # 5.
1. Let B (x, ) be a ball of radius > 0 centered at x Rn and dene
1
M f (x) = sup
|f (y )|dy.
m(B (y, ) B (y,)
>0
with the supremum taken over all balls B (y, ) such that x B (y, ).
(i) Use the covering lemmas to show that there exists a cons