exercises
and tutorials for

Functional Analysis

spring term 2010 (SoSe2010)

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.

supplementary material for the preparation towards the final test:

- Homework exercises and problems in class from the 2003/2004
Functional
Analysis
class

- Homework exercises and problems in class from the 2004/2005
Functional
Analysis
class

(IMPORTANT: the links above contain also material not covered in class this year)

final
test (Endklausur):

- Saturday 24 July 2010, 9:00-11:30,
rooms
B138, B139.

Final test July 2010 Solutions

- Saturday 6 November 2010,
9:00-11:30, rooms
C 123.

Final test November 2010 Solutions

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.