exercises and tutorials for
Functional Analysis

spring term 2012 (SoSe 2012)

(This page is not being updated)

course web page

weekly schedule:



tutorial A
tutorial B
tutorial D



tutorial C


tutorial E


session, B-005

office hours:

  Thomas Ø. Sørensen Thursday
10-11 am
office B-408
  Alessandro Michelangeli Tuesday
4-6 pm
office B-334
  Anna Zhigun Thursday
after tutorial D
  Benedikt Staffler Thursday
after tutorial E

homework assignments:
  • Each new homework assignment is posted by Tuesday evening.
  • Hand-in deadline: each Tuesday by 6pm, in the designated drop-box 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
problems in class, set 01
week 2 assignment 02 solutions problems in class, set 02
week 3 assignment 03 solutions
problems in class, set 03
week 4 assignment 04 solutions
problems in class, set 04
week 5 assignment 05 solutions
problems in class, set 05
week 6 assignment 06 solutions
problems in class, set 06
week 7 assignment 07 solutions
problems in class, set 07
week 8 assignment 08 solutions
problems in class, set 08
week 9 assignment 09 solutions
problems in class, set 09
week 10 assignment 10
problems in class, set 10
week 11 assignment 11 solutions
problems in class, set 11
week 12 assignment 12 solutions
problems in class, set 12

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)

Mon 16 July 2012, 6,30 pm, room B-052 July written test solutions
Sat 20 October 2012, 9 am, room B-052 October written test solutions

historical materials related to homework exercises (E) and problems in class (P):

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 Hahn-Banach 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 Hahn-Banach: unit ball as countable intersection of half-planes

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 Hilbert-Schmidt norm attains the operator norm when the kernel factorizes.
E42   Ehrling's lemma. L1-norm and derivative's sup control the sup norm. An Ehrling-like interpolation on gradients.
E43   Applications of Fatou's Lemma and of monotone convergence. Boundedness of the Lp-Lq 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 / non-compact subsets of Banach spaces.
E40   Examples of Lp-distances

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 first-order 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 non-separable Hilbert space of Besicovitch quasi-periodic 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 / non-compact subsets of C[0,1]
P35   Examples of compact / non-compact 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 Jordan-Von 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 sub-graph of a convex function. The orthogonal projection onto a closed ball, a hyperplane, a half-plane.
P31   Existence of algebraic basis. Non-countable 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 non-norm 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. Finite-dim 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   Norm-distance 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 non-compact 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 non-compact, compact non-Hausdorff.
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 co-countable topology.

P05   Examples of closure, interior, boundary, limit/isolated points in R. Examples of simultaneously closed and open sets.
P06   The half-open interval topology. Half-open intervals are open and closed. Continuity is the half-right continuity.
P07   Either-or topology. Partition topology. Sub-base of straight lines in the plane.
P08   Kuratowski’s closure-complement problem.

week 1   
assignment 01   problems in class, set 01

E01   Basic facts about closed, interior, boundary.
E02   The co-finite 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.