Period: Fall 2006.
Lecturers: Marcel de Jeu (first half of the course) and Andre Ran (second half of the course).
Assistant: Christian Svensson
Coordinates of Marcel de Jeu:
Visiting address: Leiden University, Niels Bohrweg 1 (Snellius building), office 218 (third floor).
Telephone: 071 527 7118.
Email: mdejeu@math.leidenuniv.nl.
Coordinates of Andre Ran:
Visiting address: Vrije Universiteit, Boelelaan, office R3.45.
Telephone: 020 598 7691.
Email: acm.ran@few.vu.nl.
Coordinates of Christian Svensson:
Visiting address: Leiden University, Niels Bohrweg 1 (Snellius building), office 211 (third floor).
Telephone: 071 527 5265.
Email: chriss@math.leidenuniv.nl.
Tutorial in Leiden
For the benefit of the participants from Leiden a tutorial accompanying this course will be offered by Marcel de Jeu in Leiden (for both the first and the second half). It goes without saying that all
participants of the national course are welcome. The tutorial is on Wednesdays from 13.45-15.30 in room 403 (second floor) of the Snellius building. The first meeting is on Wednesday September 13; due to the local schedules there will be no tutorial on October 25 and November 8. On October 11 and December 6 the tutorial can not take place due to absence of Marcel de Jeu.
Final grade
Rather than asking people to write an elaboration on some topic, we have decided to ask people to:
- study a topic on their own, chosen from a list of topics which we will offer, and
- hand in their solutions to exercises which we will choose beforehand about the topic of their choice.
The solutions to the exercises will determine 50% of the grade. The remaining 25% (the homework also accounts for 25%) will be determined on the basis of an oral discussion about the chosen topic, the exercises about this topic and, perhaps, solutions to some exercises which were part of the homework and which need improvement (if so, we will communicate the numbers of these homework exercises beforehand on an individual basis).
Tentative program
Week 1 (Sept 11): Chapter 1. Exercises: 4, 9, 10, 13, 15, 16, 22.
Week 2 (Sept 18): Chapter 2, sections 2.1-2.3; Chapter 9, section 9.5 (if time permits). Exercises: 1, 2, 9, 18.
Week 3 (Sept 25): Chapter 2, section 2.4 (sketch); Chapter 3, sections 3.1-3.3. Exercises: 19 from Chapter 2 (=M should read <=M) and 1, 4, 5 and 15 from Chapter 3.
Week 4 (Oct 2): Chapter 3 complete, Exercises: 9 (no norm considerations), 11, 13 (X should be assumed Banach here), 14 and "find an example of an operator in B(X) which is surjective and injective, but which has unbounded inverse".
Week 5 (Oct 9): Chapter 4: parts on finite dimensional spaces and equivalent norms, Riesz' Lemma, theory of metric spaces and the basics about compact operators: p. 80-84, 95-98. Exercises: 1, 5, 3+8 (solved in one go), 7. For 7, which you may find a challenging application of quotient spaces, you can use that the composition of a compact and a continuous operators is always compact again.
Week 6 (Oct 16): Chapter 4 complete. Exercises: 2, 16, 17, 18, and two additional exercises: I. Prove that a NVS is finite dimensional if and only if its norm dual X' is finite dimensional. II. Prove that the range of a compact operator between two Banach spaces can not contain a closed infinite dimensional subspace. (Remark: this is in fact a characterization of compact operators between Banach spaces).
NO LECTURE ON OCTOBER 23
Week 7 (Oct 30): Chapter 5, sections 5.1, 5.2. Exercises 2, 3, 6, 8, 9.
Week 8 (Nov 6): Chapter 5, sections 5.3, 5.4, part of 5.5 if possible. Exercises 14, 16, 17.
Week 9 (Nov 13): Chapter 6, sections 6.1-6.3. Exercises 1-5.
Week 10 (Nov 20): Chapter 6 complete. Exercises 6-10.
Week 11 (Nov 27): Chapter 7, sections 7.1, 7.2, 7.5. Exercises 2, 3, 4.
Week 12 (Dec 4): Chapter 8. Exercises 2, 4, 5, 6, 7, 20, 13.
Week 13 (Dec 11): Discussion of the projects/assignments
Week 14 (Dec 18): Chapter 9, sections 9.1-9.5. Exercises: see under Comments/reading guide below.
Week >= 15: Assignment for chosen topic and oral exam.
Assignments
First assignment, to be handed in October 2
Chapter 1: 3
Chapter 2: 14, 22. In 22 you may assume that M is closed and you are advised to look at Theorem 2.1.
Second assignment, to be handed in October 16
Chapter 3: 17, 25, 28.
Third assignment, to be handed in November 6
Chapter 4: 13, 20, 21. For 21 you may have a look at page 87 for inspiration.
Fourth assignment, to be handed in November 27
Chapter 5: 14 and 16. (Note: these numbers as well as the deadline are different from previous announcements.)
Fifth assignment, to be handed in December 4
Chapter 6: 9 and 10.
Sixth assignment, to be handed in December 18
Chapter 8: 6 and 10.
Seventh assignment, to be handed in January 15 by regular mail or email to Christian Svensson:
Chapter 9: 1 and 2. Exercise 2 is concerned with what is known in the literature as topological zero-divisors.
Comments/reading guide
Week 1
We covered Chapter 1. You need not pay attention to the details of the variation of constants (or parameters) in section 1.1.
The main point is that the initial value problem is equivalent to (1.7). There are many ideas and details in section 1.1; rest assured that
the basic notions will be covered at a more leisurely pace later on. Think of section 1.1 as a high-powered advertisement for the functional analytic approach, and not as something
which you are supposed to fully grasp at this point of the course.
If you are not familiar with Lebesgue theory, read the material in section 1.4 on Fourier series in a loose manner. It is used implicitly
that the metric completion of the continuously differentiable functions is L_2[0,2\pi], but this is not trivial. Likewise, to fully understand the material on Fourier analysis you
would need to know that the functions \phi_n are indeed an orthonormal basis of L_2[0,2\pi], which is also not trivial. Concentrate on the general material on Hilbert spaces in this section,
regarding the Fourier part as a motivating application.
If you don't like the definition of the completion of a normed linear space on page 20, you can simply accept that such an object exists. We will get it for free later on.
The most relevant exercises are 4 (important technique of proof), 10, 13 (creativity required) and 22. Sketch of solutions: Chapter 1 page 1, Chapter 1 page 2.
Week 2
Covering sections 2.1-2.3 we concentrated on the dual space of a NVS. The first basic questions is: if the space is nonzero, is the dual nonzero? The answer is affirmative, as we see from Theorem 2.7: this follows from the Hahn-Banach theorem. This HB-theorem, together with results such as Theorem 2.7 and Corollary 2.8, is one of the cornerstones of functional analysis. The second basic question is: given a concrete NVS X, can we describe its dual concretely? For Hilbert spaces this is relatively easy: H can be identified with its dual, cf. Theorem 2.1. The proof of this is based on the fundamental decomposition result Theorem 2.3, which in turn is based on the geometric Theorem 2.2. For some important other classes of spaces a description of the dual is also known, we will see this in week 3.
If you are not familiar with Zorn's Lemma, this is the time to become acquainted with it. It is an important tool and not only in functional analysis. Read section 9.5 (but not the proof of Theorem 9.11 contained in it) and convince yourself that there is really hardly anything to it. Once one know that Principle 9.17 follows from the Axion of Choice, one has a very powerful and easy-to-use tool at one's disposal - provided one accepts the Axiom of Choice of course.
For exercise 18 you will need Theorem 2.7; perhaps it is even better to reformulate this exercise as "show that a NVS can be isometrically embedded into the Banach space of all bounded continuous functions on the unit sphere in its dual". This latter space is actually a Banach algebra (with pointwise multiplication), so that every NVS (which need not have any multiplicative structure) can be isometrically embedded into a Banach algebra. Note that HB is crucial in proving this!
Sketch of solutions: Chapter 2, page 1
Week 3
We gave an extended version of section 2.4, presenting material on (the dual of) l_p-spaces, L_p-spaces and spaces C_0(X). The statement that the dual of C_0(X) can be described in terms of regular Borel measures is the famous Riesz Representation Theorem. For intervals on the real line, such a measure is always given by a function of bounded variation, so that Theorem 2.14 results (which can be proved in this special case without Riesz' result, as done in the book).
In Section 3.1-3.3 we concentrated on bounded operators and their adjoints. There are several a priori results known about the relationship between an operator and its adjoint, so that studying the adjoint in a given situation can give substantial information about the operator itself. These a priori relations are usually formulated in terms of (or can be derived using) annihilators of kernels and ranges. Equation 3.13 and Theorem 3.7 illustrate this. One of the highlights in this direction is the statement that, for a bounded operator A between two Banach spaces X and Y, the operator A (resp. A') is surjective precisely when the operator A' (resp. A) is a homeomorphism between Y' (resp. X) and its image in X' (resp. Y). If you want to be impressed by what is known in this direction, look up the so-called "state diagram" for an operator and its adjoint in "Introduction to Functional Analysis" by Taylor and Lay, or in Goldberg's "Unbounded Linear Operators" if you want to convince yourself that this can even be done in the unbounded case.
Sketch of solutions: Chapter 3, page 1, Chapter 3, page 2.
Week 4
We covered Sections 3.4-3.7, which can be described as "consequences of completeness". The basis for these results is the Baire Category Theorem 3.9 for complete metric spaces and the important Banach-Steinhaus Theorem 3.17 (uniform boundedness principle) is an almost immediate consequence of Baire's result. Somewhat more work is necessary to obtain the conglomerate of the Closed Graph Theorem (3.10), Bounded Inverse Theorem (3.8) and Open Mapping Theorem (3.18).
As an application we obtained results about closed ranges of operators (Theorem 3.14) and Theorem 3.16, the latter being one of the standard "a priori relations" between an operator and its adjoint.
In the passing we developed the useful tool of quotient spaces modulo a closed subspaces. The moral is that the algebraic structure of X/M and its quotient topology are compatible in the sense that there is a norm (the distance of any representative to M) on the quotient which gives precisely the quotient topology: the quotient topology is normable. Moreover, factor maps (which exist by general principles from topology) are then bounded linear maps with the same norm as the original map. The canonical map from X to X/M is open. Furthermore. X is Banach if and only if X and X/M are Banach.
Many of the results in this chapter are derived from closed operators with a domain that can be a proper subspace. It won't harm the understanding to assume that the domain is the entire space while reading it at this stage. The reason that it is done in this generality is the future application to unbounded operators: many differential operators, which are unbounded, have closed extensions and a theory which applies to closed operators therefore gives information about these unbounded operators which are of obvious practical interest.
Now that this chapter is finished we have the three basis principles of functional analysis at our disposal: the Hahn-Banach theorems, the Banach-Steinhaus theorem (uniform boundedness principle) and the Closed Graph Theorem/Bounded Inverse Theorem/Open Mapping Theorem.
Remark: the proof of Theorem 3.18 is not correct, since D is Banach if and only if D(A) is closed, which is not a part of the hypotheses. The theorem is correct however, and the proof can be found as item II.1.8 in Goldberg's "Unbounded linear operators".
Sketch of solutions: Chapter 3, page 3, Chapter 3, page 4.
Week 5
We embarked on developing the theory of compact operators in Chapter 4. This chapter is only part of the story: Theorem 6.2 gives important information about the possible eigenvalues and Chapter 5 (Fredholm theory) contains further generalizations of the results in the current chapter.
Basic facts about finite dimensional spaces precede the theory of compact operators: all norms on a finite dimensional space are equivalent, the space is always complete, finite dimensional subspaces of a NVS are always closed. Based on the Riesz Lemma 4.7 one shows that finite dimensionality of a NVS, the validity of the Heine-Borel theorem ("compact = closed and bounded"), the compactness of the closed unit ball, the compactness of the unit sphere and the compactness of the identity map are all equivalent.
It is an important fact that for a compact operator T in B(X,Y) its adjoint T' is again a compact operator from Y' to X'. The converse is also true if Y' is Banach, as is easily derived from the previous fact using the canonical isometrical embedding of any NVS into its second norm dual (see Chapter 8). In order to understand this conclusion about T' we proved basic results about totally bounded and relatively compact subsets of metric spaces.
Sketch of solutions: Chapter 4, page 1, Chapter 4, page 2.
Week 14
We covered the Sections 9.1-9.5 where the basic theory of Banach algebras is presented. Prototypical unital Banach algebras are B(X) for X a Banach space, C(X) for X a compact topological space and L_1(G) where G is a discrete topological group. These three types stand for algebras of operators, algebras of functions and convolution algebras on groups.
In these sections you will recognize many results from earlier chapters about B(X); these are simply the results the proofs of which used only the fact that B(X) is a Banach algebra. Note the use of complex function theory in the proof of Theorem 9.6 (non-triviality of the spectrum); it is this combination of complex function theory, algebra and the theory of normed vector spaces which makes the theory of Banach algebra rather appealing to many people.
We went a bit further than the book in Section 9.2 where the Banach algebra interpretation of Fredholm theory is discussed. The complement of \Phi_A is simply the spectrum of the class of A in the Banach algebra B(X)/K(X), which is called the Calkin algebra of X. The index map descends to the invertible elements in the Calkin algebra (since the index does not change under compact perturbations) and then yields a group homomorphism of this group of invertible elements into the integers (since i(AB)=i(A) + i(B) for Fredholm operators A and B. If X is a Hilbert space, then this homomorphism is surjective and the kernel can be shown to be the connected component of the identity element in the group of invertible elements in the Calkin algebra. All in all, we conclude that in the Hilbert space setting the (descended) index maps labels the connected components of the group of invertible elements in the Calkin algebra.
The proof of Theorem 9.12 in the book is not correct. It uses that each maximal ideal is of codimension 1 in B and then proceeds to show that it is trivially the kernel of a multiplicative linear functional. It is, however, not proved that the codimension must be one. In fact, the correct proof is just the other way round. If N is a maximal ideal, then B/N is a field and it follows easily from Theorem 9.6 that any Banach algebra which is a field is isomorphic to the complex numbers (this is known as the Gelfand-Mazur theorem). Composing this isomorphism with the natural map of B into B/N we find a multiplicative linear functional whose kernel is precisely N, as desired. Is is only then that we can conclude that N has codimension 1.
Exercises and solutions: Chapter 9, page 1, Chapter 9, page 2, Chapter 9, page 3.
