exercises
and tutorials for
Functional Analysis
spring term 2010 (SoSe2010)
(This page is not
being updated)
course
web page
HOME
homework
assignments:
- Each new homework assignment is posted by Monday afternoon.
- Hand-in deadline: each Monday by 12 pm, in the
designated drop-box
located on the first floor.
- Pick up your marked worksheets from the designated return box on
the first floor.
- A sketch
of the solution to each exercise will be posted weekly.
- A selection of
the homework exercises will be discussed in detail during
the homework session.
- Please
register
through the LMU Maths Institute homework
webpage.
problems
in
class:
- Problems in class are additional problems are for
your own preparation at home.
- They supplement examples and properties
not discussed in class.
- Some of them will discussed interactively in
the weekly exercise/tutorial sessions.
- You are not required to hand in
their solution. You are encouraged to think them over and to solve
them.
| week 1 |
assignment
01
|
solutions
|
problems
in class, set 01
|
solutions
|
| week 2 |
assignment
02 |
solutions
|
problems in class, set 02
|
solutions
|
| week 3 |
assignment
03 |
solutions
|
problems in class, set 03
|
solutions
|
| week 4 |
assignment
04 |
solutions
|
problems in class, set 04
|
solutions
|
| week 5 |
assignment
05 |
solutions
|
problems in class, set 05
|
solutions
|
| week 6 |
assignment
06 |
solutions
|
problems in class, set 06
|
solutions
|
| week 7 |
assignment
07 |
solutions
|
problems in class, set 07
|
solutions
|
| week 8 |
assignment
08 |
solutions
|
problems in class, set 08
|
solutions
|
| week 9 |
assignment
09 |
solutions
|
problems in class, set 09
|
solutions
|
| week 10 |
assignment
10
|
solutions
|
problems in class, set 10
|
solutions
|
| week 11 |
assignment
11 |
solutions
|
problems in class, set 11
|
solutions
|
supplementary
material for the preparation towards the final test:
final
test (Endklausur):
markers:
markers
|
office
hours
|
Pablo
|
Mon 14-16,
A412
|
José
|
Tue 18-19,
B404
|
Martin
|
on request
(nach Vereinbarung)
|
Dietmar
|
on request
(nach Vereinbarung) |
exercise/tutorial
sessions:
day
|
time
|
lecture
hall |
tutor |
Monday
|
10-12 am
|
B134
|
José |
| Wednesday |
4-6 pm |
C113 |
Ale |
| Thursday |
4-6 pm |
B051 |
Ale |
CREDITS:
Some
of
the
homework
exercises
or
problems in class contain classical
results from the past or from recent research papers.
- Exercise 7 uses the uniform boundedness of coefficients of
polynomials up to degree N, a result that in a wider generality was
proved first by O. Szász in Ungleichungen
für die Koeffizienten einer Potenzreihe, Math. Z. 1
(1918) 163-183.
- Exercise 16 deals with the uniform approximation of continuous
functions with Bernstein polynomials that was first proved by S.
Bernstein in Démonstration du
théorème de Weiestrass, fondeé sur le calcul des
probabilités, Comm. Soc. Math. Kharkow (2), 13 (1912), 1-2. For a comprehensive
treatment of Bernstein polynomials, see G. G. Lorenz, Bernstein
Polynomials, University of Toronto Press, Toronto, Canada,
1953.
- Exercise 17 contains Hardy's inequality, first proved by G. H.
Hardy in Note
on
a
theorem
of
Hilbert, Math. Zeitschr. 6
(1920), 314-317.
- Exercise 18 (iii) is just estimate (2.37) from A. Michelangeli,
B. Schlein, Dynamical collapse of Boson stars,
Comm.
Math. Phys. 311 (2012),
645-687
- Exercise 20 (ii) (a normed space has a inner product iff the
parallelogram law
holds true) is classical result by Jordan and von Neumann
in On inner
products in linear, metric spaces (1935).
The
algebraic formulation of the parallelogram identity appears
in Proposition
5
of
Book
II
of
Euclid's
Elements. You can read it in one of
the oldest fragments of papyrus today available of Euclid's
Elements.
- Exercise 21is one of the many problems on "quadratically close
ONB" aka
"completeness of bi-orthogonal systems". The first one is apparently
due to Paley and Wiener, Fourier
transforms in the complex domain (1935), page 100. That result is
later discussed in the classical monography by Riesz and Nagy, Functional
Analysis (1952), paragraph 86. The version discussed here in E35 is
apparently due first to Bari (Nina Karlovna
Bari) in her article Biorthogonal systems and bases in
Hilbert
spaces (1951) and reproved later by Birkhoff and Rota in On the completeness
of Sturm-Liouville expansions (1960), by Urata in A
theorem
of
Bari
on
the
completeness
of
orthonormal
system (1963),
and by Tsao in Approximate
bases in a Hilbert space (1968). For a comprehensive treatment of
quadratically close bases, see chapter VI of Gohberg and Krein, Introduction
to
the
theory
of
linear nonselfadjoint operators (1969).
- Last, Cauchy-Schwarz's inequality is of course neither Cauchy's
nor
Schwarz's, but Bunyakowsky's. "Cauchy-Schwarz" seems to have become
standard
only since the 1930s. The first JSTOR match is in 1930 -- an
article by A. E. Ingham -- and the term appears in the
widely-used Differential and Integral Calculus, 2nd. ed. by R.
Courant (1937).
The history of the contributing inequalities is given in Inequalities
by G. H. Hardy, J. E.
Littlewood and G. Polya (1934): the inequality for sums is due to A.
L.
Cauchy
in 1821 (p. 373 of Oeuvres
2, III)
and the inequality for integrals to H.
A.
Schwarz in 1885, "although it seems to have been stated first by Buniakovsky"
in
1859.
In
Russia
the
integral
version
is known as the Bunyakovsky
inequality. Bunyakowsky's original paper is Sur quelques inegalités concernant
les intégrales aux différences finis, Mem.
Acad. Sci. St. Petersbourg (7) 1
(1859)
pp. 9.
