MATH 7338
HOMEWORK #1
Due date: September 22, 2009
Work the following problems and hand in your solutions. You may work together with
other people in the class, but you must each write up your solutions independently. A subset
of these will be selected fo
MATH 7338
HOMEWORK #2
Due date: October 8, 2009
Work the following problems and hand in your solutions. You may work together with
other people in the class, but you must each write up your solutions independently. A subset
of these will be selected for g
MATH 7338
HOMEWORK #3
Due date: October 29, 2009
Work FIVE of the following problems and hand in your solutions. You may work together
with other people in the class, but you must each write up your solutions independently. A
subset of these will be selec
MATH 7338
HOMEWORK #4
Due date: December 3, 2009
Work FOUR of the following problems and hand in your solutions. You may work together
with other people in the class, but you must each write up your solutions independently. A
subset of these will be selec
FUNCTIONAL ANALYSIS LECTURE NOTES:
EGOROFF AND LUSINS THEOREMS
CHRISTOPHER HEIL
1. Egoroffs Theorem
Egoros Theorem is a useful fact that applies to general bounded positive measures.
Theorem 1 (Egoros Theorem). Suppose that is a nite measure on a measure
SOME RELEVANT AND NOT-SO-RELEVANT TEXTS
Christopher Heil
School of Mathematics, Georgia Tech
This is a sampling of some texts in subjects more-or-less related to the courses I usually teach, limited to books
that I happen to know and/or like.
1. Hilbert S
REVIEW OF LEBESGUE MEASURE AND INTEGRATION
CHRISTOPHER HEIL
These notes will briey review some basic concepts related to the theory of Lebesgue
measure and the Lebesgue integral. We are not trying to give a complete development,
but rather review the basi
14
1 The Fourier Transform on L1 (R)
1.8. This problem provides an alternative proof to Theorem 1.17.
(a) Show that f C0 (R) for every f S = spancfw_[a,b] : a < b R.
(b) Show that S is dense in L1 (R) (see Exercise B.61), and use this to
prove that f C0 (
FUNCTIONAL ANALYSIS LECTURE NOTES: PROBLEMS ON c AND c0
CHRISTOPHER HEIL
Definition 1. Sequences with unspecified limits are indexed by the natural numbers N. We set c = c(N) = c0 = c0 (N) = c00 = c00 (N) = a = (ak ) : lim ak exists ,
k
a = (ak ) : lim ak
FUNCTIONAL ANALYSIS LECTURE NOTES: COMPACT SETS AND FINITE-DIMENSIONAL SPACES
CHRISTOPHER HEIL
1. Compact Sets Denition 1.1 (Compact and Totally Bounded Sets). Let X be a metric space, and let E X be given. (a) We say that E is compact if every open cover
REAL ANALYSIS LECTURE NOTES: 2.4 MODES OF CONVERGENCE
CHRISTOPHER HEIL
2.4.1 The relation between convergence in measure and pointwise convergence Although convergence in measure does not imply pointwise convergence, we do have the following weaker (but s
A
Metrics, Norms, Inner Products, and Topology
These appendices collect the background material needed for the main part of
the volume. In keeping with the philosophy of this text, we formulate these as
mini-courses with the goal of providing substantial,
Exercises from Appendix A
N
A.12 Hints: . Let sN = n=1 fn , and show that the sequence of partial
sums cfw_sN N N is Cauchy in X.
. Suppose that every absolutely convergent series is convergent. Let
cfw_fn nN be a Cauchy sequence in X. Show that there exi
C Functional Analysis and Operator Theory
C.1 Linear Operators on Normed Spaces
In this section we will review the basic properties of linear operators on normed spaces. Definition C.1 (Notation for Operators). Let X, Y be vector spaces, and let T : X Y b
Exercises from Appendix C
C.10 Hint: Expand f + g
using the Polar Identity.
2
= Lf + Lg
2
and f + ig
2
= Lf + iLg
2
C.14 Hint: (b) If , then there exists a subsequence (nk )kN such that
/
|nk | k for each k . Let cnk = 1/k and dene all other cn to be zero
D
Borel and Radon Measures on the Real Line
In this appendix we review the theory of signed and complex Borel and Radon
measures. These can be dened on locally compact Hausdor spaces, but for
the purposes of this volume we only need to deal with Borel and
E
Topological Vector Spaces
Many of the important vector spaces in analysis have topologies that are
generated by a family of seminorms instead of a given metric or a norm.
We will consider these types of topologies in this appendix. References for
the ma
F Complex Analysis
In this appendix, we collect a few basic definitions and facts from complex analysis. Complex analysis is a vast and beautiful subject. Although we make only limited use of complex analysis in this volume, there is a rich interaction be