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 solutio
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
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 s
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 s
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
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 t
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
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)
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
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
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 pointw
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
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
c
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 O
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
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 purp
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 type
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