weekly schedule:

Monday

Tuesday

Wednesday

Thursday

Friday

810






1012


tutorial A
B045

tutorial B
B039

tutorial D
B045


1214






1416

lecture
B005


tutorial C
B045



1618



lecture
B005

tutorial E
B133


1820


homework
session, B005




office hours:
Thomas Ø. Sørensen 
Thursday

1011 am

office B408 
Alessandro Michelangeli 
Tuesday

46 pm

office B334

Anna Zhigun 
Thursday

after tutorial
D 

Benedikt Staffler 
Thursday

after tutorial
E 

homework assignments:
 Each new homework assignment is posted by Tuesday
evening.
 Handin deadline: each Tuesday by 6pm, in the designated
dropbox located on the first floor.
 Pick up your marked worksheets from the designated
return box on the first floor.
 Markers: Markus Furtner, Michael Handrek, Jan Priel
(contact them for clarifications on marking).
 A sketch of the solution to each exercise will be posted
weekly.
 A selection of the homework exercises will be discussed
in detail in the homework session.
 You may find the diary of homework sessions and
tutorials at the bottom of this page.
 Please register through the LMU Maths Institute homework
webpage.
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

week 12 
assignment 12 
solutions

problems in
class, set 12

solutions

supplementary problems
for the preparation towards the final test
(notice, though, that they cover also topics NOT included in
the present course):
final test (Endklausur)
historical materials related
to homework exercises (E) and problems in class (P):
 The
emergence
of open sets, closed sets, and limit points in analysis
and topology
 Some
remarks
on the interaction of general topology with other areas
of mathematics
 P08 is the classical Kuratowski's closurecomplement
problem  see the original
paper (1922), English
version here.
 Venn's
diagram stained glass window memorial in the
Gonville and Caius college dining hall at Cambridge
 1914 original edition of Hausdorff's Grundzüge
der Mengenlehre
 Banach's
1922 article (condensing his 1920 PhD thesis) with
the fixed point theorem, the def. of Banach space, etc.
 Elliott's
1926 article with his proof of Hardy's inequality,
needed in E25(i)
 On Hardy's inequality, see also: The
Prehistory of the Hardy Inequality.
 Euler's 1740 celebrated article De
summis series reciprocarum (English version here)
solving the Basel
problem. Needed in E26(ii).
 Minkowski's Geometrie
der
Zahlen (1910): triangular inequality in l^p
(discussed in P25(i)) is in Chapter 3, paragraph 30.
 In P27(ii) Ptolemaic
inequality is discussed for general inner product
spaces. In Euclidean geometry this result was established
in Ptolemy's
Almagestum (the link is to a Latin edition of 1515),
precisely in book
1,
pag. 6.
 E30 (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.
 E33 (every separable Banach space is a quotient of l^1)
is a classical result by Banach and Mazur in Zur
Theorie der linearen Dimension (1933)
 In E34 the nonseparable Hilbert space of Besicovitch
quasiperiodic functions is discussed, which was
introduced first by Besicovitch in On
mean
values of functions of a complex and of a real variable
(1927) and On
Parseval's
theorem for Dirichlet series (1927)
 E35 is one of the many problems on "quadratically close
ONB" aka "completeness of biorthogonal 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 SturmLiouville 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).
 Schur's test, discussed in E36(i), was introduced by
Schur in Bemerkungen
zur Theorie der beschränkten Bilinearformen mit
unendlich vielen Veränderlichen (1911)
 Hilbert's matrix (i.e., whose entries are unit
fractions) was introduced by Hilbert in Ein
Beitrag
zur Theorie des Legendre'schen Polynoms (1894). The
problem of estimating the norm of its infinite dimensional
analogue, which is discussed in E36(ii), originated the
socalled Hilbert's double series theorem and was proved
first (apart from the exact determination of the constant)
by Hilbert in his lectures on integral equations.
Hilbert's proof was published by Weyl in his PhD thesis Singuläre
Integralgleichungen mit besonderer Berücksichtigung des
Fourierschen Integraltheorems (1908). The
determination of the constant (and the integral analogue)
is due to Schur in the article Bemerkungen
zur Theorie der beschränkten Bilinearformen mit
unendlich vielen Veränderlichen (1911). The
extension from l^2 to general l^p spaces is due to Hardy
and Riesz  see Hardy's article "Note on a theorem of
Hilbert concerning series of positive terms" in Proc.
London Math. Soc. (2), 23 (1925).
 The notion of equicontinuity was introduced at around
the same time by Giulio
Ascoli (1883–1884) and Cesare
Arzelà (1882–1883). Ascoli established first the
sufficient condition for compactness. Arzelà (1895)
established the necessary condition and gave the first
clear presentation of the result. A more expanded history
of AscoliArzelà theorem
 E38(i) (the Cesaro
means of the partial sum of the Fourier series of f
converge uniformly to f) is a result first established by
Fejér
(about him, see also here)
in Untersuchungen
über Fouriersche Reihen (1903). Cesaro's
summation method was introduced by Cesaro in Sur la
multiplication des séries," Bulletin des Sciences
Mathématiques, 14, 114120 (1890).
 In P39(i) we discuss Peano's existence theorem of a
local solution to the initial value problem of a first
order ODE (continuity is the only assumption), which was
established first by Peano in Démonstration
de l'intégrabilité des équations différentielles
ordinaires (1890).
 In P39(ii) we discuss Picard–Lindelöf existence and
uniqueness theorem of a local solution to the initial
value problem of a first order ODE (Lipschitz continuity
is needed), which was established by Picard in Sur
l'application
des méthodes d'approximations successives à l'étude de
certaines équations différentielles ordinaires
(1893) and soon after generalized by Lindelöf in Sur
l'application
de la méthode des approximations successives aux
équations différentielles ordinaires du premier ordre
(1894).
 In E42 Ehrling's lemma is discussed, which was proved
first by Ehrling in On a
type of eigenvalue problems for certain elliptic
differential operators (1954).
diary of homework assignments and tutorials (table of
contents):
(E=Exercise, P=Problem in class)
*******************************************
week 12
assignment 12 problems in class,
set 12
E45 An application of Baire and Closed Graph
E46 The Cantor lemma in Banach spaces
E47 Extension of bounded linear functionals under
further constraints
E48 Explicit calculation of all the HahnBanach
extensions
P45 L^2 as a first category set that is not nowhere
dense
P46 An application of the Uniform Boundedness Principle
P47 An application of the Closed Graph theorem:
projections on Banach spaces
P48 Consequence of HahnBanach: unit ball as countable
intersection of halfplanes
*******************************************
week 11
assignment 11 problems in class,
set 11
E41 Integral operators on C[0,1] and L^p[0,1].
Boundedness and compactness. The HilbertSchmidt norm attains
the operator norm when the kernel factorizes.
E42 Ehrling's lemma. L1norm and derivative's sup
control the sup norm. An Ehrlinglike interpolation on
gradients.
E43 Applications of Fatou's Lemma and of monotone
convergence. Boundedness of the LpLq multiplication operator.
E44 Orthonormal bases on an interval
P41 Regularity of Fourier series
P42 Sobolev spaces on [0,2pi]
P43 Computation of weak derivatives. Weak
differentiation and integration commute.
P44 Commutativity of weak derivatives. Multiplication
by a C^inf_0 function leaves Sobolev spaces invariant.
*******************************************
week 10
assignment 10 problems in
class, set 10
E37 A normalized system of vectors with fixed angle
among them converge "weakly''
E38 Cesaro summability. The canonical ONB of L^2[0,1]
E39 Examples of compact / noncompact subsets of Banach
spaces.
E40 Examples of Lpdistances
P37 A continuity property for a sequence of orthogonal
projections
P38 An orthogonal projection on L^2(R+)
P39 Solutions to the initial value problem of a
firstorder ODE: existence and uniqueness à la Peano and à la
Picard.
P40 Smooth functions are not dense in Hölder continuous
functions
*******************************************
week 9
assignment 09 problems in class,
set 09
E33 Every separable Banach space is a quotient of l^1
E34 The nonseparable Hilbert space of Besicovitch
quasiperiodic functions
E35 Quadratically close orthonormal bases
E36 Schur's test. Hilbert's matrix. Hankel's matrix.
P33 An orthogonal projection on L^2[1,1]
P34 Examples of compact / noncompact subsets of C[0,1]
P35 Examples of compact / noncompact operators on a
Banach space
P36 Inclusion of spaces reverts for their dual if one
space is dense in the other. Variational characterisation of
the norm of an operator on a Hilbert space.
*******************************************
week 8
assignment 08 problems in class,
set 08
E29 Another Hardy’s operator on l^p.
E30 A normed space has a inner product iff the
parallelogram law holds true (the JordanVon Neumann theorem).
E31 C([0,1]) embeds isometrically into l^\infty, not
into l^p if p is finite.
E32 c and c_0 are not isomorphic (as Banach spaces) but
their duals are.
P29 Projections onto closed convex sets. Generalisation
of the orthogonal projection onto a closed subspace.
P30 The orthogonal projection onto a subgraph of a
convex function. The orthogonal projection onto a closed ball,
a hyperplane, a halfplane.
P31 Existence of algebraic basis. Noncountable bases.
P32 Existence of unbounded linear functionals. They
have dense kernel.
*******************************************
week 7
assignment 07 problems in class,
set 07
E25 Hardy’s operator on l^p
E26 Computation of norm of functionals.
E27 Distance from a point to a set in a normed space.
E28 Norm attaining and nonnorm attaining bounded
linear functionals.
P25 Equality in Minkowski’s inequality. Variational
characterization of norm in l^p spaces and in Hilbert spaces.
P26 Algebraic equation for compact operators. The l^p
spaces are nested. They don't embed into each other compactly.
P27 Ptolemaic inequality. The unit ball is strictly
convex in Hilbert spaces.
P28 Constructing new Banach spaces as quotients of
Banach spaces.
*******************************************
week 6
assignment 06 problems in class,
set 06
E21 Completeness = nested balls shrink to a point.
Examples of completions.
E22 Norms are 1:1 with translation invariant,
homogeneous metrics. Triangular inequality for norms iff
closed unit ball is convex. Finitedim norms are equivalent.
E23 Banach is the same as absolutely convergent
sequences converge. Application of the contraction principle
to Banach spaces.
E24 Closure of l^p in l^q. Separability of l^p.
P21 Computation of operator norms.
P22 The value attained by a functional at a point is
determined by distance from kernel. Characterization of when a
functional attains its norm.
P23 Normdistance between hyperplanes.
P24 Dual of c_0. Dual of l^1. Dual of l^infty.
*******************************************
week 5
assignment 05 problems in class,
set 05
E17 Contractions on compact metric spaces. No
surjective contractions on a compact. Iterated maps that are
contractions. An inverse of Banach fixed point.
E18 A noncompact unit ball. Examples of open, closure,
interior in C[0,1].
E19 Topologies with the same convergent sequences.
Equivalent metrics. Equivalent norms.
E20 Examples of open/closed in l^p.
P17 A continuous function on XxY is continuous in each
variable, not vice versa. Closure and interior of a product.
P18 When E+F is open or closed or compact in a normed
space.
P19 Completion is closure, if the ambient space is
complete. Compactness = complete + totally bounded.
P20 The closure of c_00 in l^p. When the closure of an
open ball is the closed ball in generic metric spaces. Always
true in normed spaces.
Fact recalled in class:
* Examples of metric spaces, homeomorphic or just with the
same topology, but one is complete whereas the other is not:
E15(i), E15(iii), E19(ii).
*******************************************
week 4
assignment 04 problems in class,
set 04
E13 Open covers with no finite subcover. Closed+bounded
is less then compact, if space not complete. Hausdorff
noncompact, compact nonHausdorff.
E14 Continuous bijections are not homeomorphisms in
general. They are between compact Hausdorff spaces.
E15 Homeomorphisms do not preserve completeness. A
metric making irrationals complete and homeomorphic to the
Euclidean irrationals.
E16 R is isometrically homeomorphic to the punctured
disk, its completion is the disk.
P13 Cauchy and fast Cauchy. A metric is uniformly cont.
in each entry. Continuous functions between metric spaces are
uniformly cont. if domain is compact.
P14 Subspaces of separable metric spaces are separable.
Lindelöf's theorem. For metric spaces compact <=>
sequentially compact.
P15 The completion theorem in full detail. Uniqueness
of the completion up to isometry.
P16 Contractions with no fixed points. Compact implies
bounded. Product of compacts is compact. Compactness = finite
intersection property.
*******************************************
week 3
assignment 03 problems in class,
set 03
P11 is discussed also in
Problem
6 here and
Problem
9 here.
E09 Hausdorff topological spaces.
E10 Second countable implies separable, not vice versa.
E11 Finite product topology. Projections are cont and
open. Universal property. Product is Hausdorff iff factors
are. The diagonal.
E12 A metric on R making each rational an open set.
P09 Finite product topology: separability,
countability, closed graph.
P10 The weak topology: base, universal property.
Product topology and product convergence. Pointwise
convergence topology.
P11 Examples of metrics in R^d and in C[0,1]
P12 Topological and uniform equivalence of metrics. d1
and d_inf are not equivalent, nor the topologies are one into
the other.
Facts recalled in class:
* P01(i), P01(ii), P05(ii), P05(iii), P06(i)(ii), P07(ii)
provide a zoo of examples of sets that are both open and
closed!
* Remark 1.24(2) stated in class, and deferred to tutorials,
is discussed in P09(ii)
* Remark 1.24(3) stated in class, and deferred to tutorials,
is discussed in P10(v)
*******************************************
week 2
assignment 02 problems in class,
set 02
E05 Elementary facts on closure, interior, boundary
with respect to complement, inclusion, union, intersection.
E06 Base for a topology. Characterisation and relation
with a basis of neighbourhoods.
E07 Characterisation of closure and closed sets. {Limit
points} is closed in a Housdorff topology.
E08 Limits of sequences do not identify the closure, in
general. The case of the cocountable topology.
P05 Examples of closure, interior, boundary,
limit/isolated points in R. Examples of simultaneously closed
and open sets.
P06 The halfopen interval topology. Halfopen
intervals are open and closed. Continuity is the halfright
continuity.
P07 Eitheror topology. Partition topology. Subbase of
straight lines in the plane.
P08 Kuratowski’s closurecomplement problem.
*******************************************
week 1
assignment 01
problems in class,
set 01
E01 Basic facts about closed, interior, boundary.
E02 The cofinite topology: closure of a set, all
convergent sequences.
E03 Relative topology: relatively closed, relative
closure, transitivity.
E04 Relative topology: relative closure in
general, relative convergence.
P01 Closure and continuity in the discrete and
indiscrete topologies.
P02 Openness and closedness in the relative
topology.
P03 Kuratowski’s closure axioms.
P04 Simple examples of homeomorphic topological
spaces.
*******************************************