Preface
These notes are for a one-semester graduate course in Functional Analysis,
which is based on measure theory. The notes correspond to the course
Real Analysis
II
, which the author taught at University of Michigan in the Fall 2010. The course
consists of about 40 lectures 50 minutes each.
The student is assumed to be familiar with measure theory (both Lebesgue and
abstract), have a good command of basic real analysis (epsilon-delta) and abstract
linear algebra (linear spaces and transformations).
The course develops the theory of Banach and Hilbert spaces and bounded
linear operators.
Main principles of are covered in depth, which include Hahn-
Banach theorem, open mapping theorem, closed graph theorem, principle of uni-
form boundedness, and Banach-Alaoglu theorem. Fourier series are developed for
general orthogonal systems in Hilbert spaces.
Compact operators and basics of
Fredholm theory are covered.
Spectral theory for bounded operators is studied in the second half of the course.
This includes the spectral theory for compact self-adjoint operators, functional
calculus and basic spectral theory of general (non-compact) operators, although
the latter needs to be expanded a bit.
Topics not covered include: Krein-Milman theorem (although this can be done
with one extra lecture), unbounded linear operators, and Fourier transform. Most
applications to ODE and PDE are not covered, however the integral operators serve
as a main example of operators in this course.
The material has been compiled from several textbooks, including Eidelman,
Milman and Tsolomitis “Functional Analysis”, Kirillov and Gvishiani “Theorems
and problemsin functional analysis”, Reed and Simon “Methods of modern mathe-
matical physics. I. Functional analysis”, V. Kadets “A course in functional analy-
sis” (Russian), and P. Knyazev, “Functional analysis”. Minor borrowings are made
from Yoshida “Functional analysis”, Rudin “Functional analysis”, and Conway “A
course in functional analysis”. For some topics not covered, one may try R. Zimmer
“Essential results of functional analysis”.
Acknowledgement.
The author is grateful to his students in the Math 602
course
Real Analysis II
, Winter 2010, who suggested numerous corrections for these
notes. Special thanks are to Matthew Masarik for his numerous thoughtful remarks,
corrections, and suggestions, which improved the presentation of this material.