Venue: Buys Ballot building, Utrecht University, room 061.
Time and dates of lectures: Tuesdays from 10.15-13.00 on
September 6, 13, 20, 27, October 4, 11, 25, November 1, 8, 15, 22, 29, and
December 6, 13 (i.e., all Tuesdays in the weeks 36 through 50, with
a break in week 42).
Coordinates of Marcel de Jeu
Visiting address: Mathematical Institute, Leiden University, Niels Bohrweg 1 (Snellius building), office 218 (third floor).
Telephone: 071 527 7118.
Email: mdejeu@math.leidenuniv.nl.
Postal address:
Mathematical Institute
Leiden University
P.O. Box 9512
2300 RA Leiden
Coordinates of Andre Ran
Visiting address: Vrije Universiteit, De Boelelaan 1081, office R3.45.
Telephone: 020 598 7691.
Email: acm.ran@few.vu.nl.
Postal address:
Afdeling Wiskunde
Vrije Universiteit
Faculteit der Exacte Wetenschappen
De Boelelaan 1081a
1081 HV Amsterdam
Coordinates of Blaz Mramor
Visiting address: Vrije Universiteit, De Boelelaan 1081.
Telephone: .
Email: bmramor@few.vu.nl.
Postal address:
Afdeling Wiskunde
Vrije Universiteit
Faculteit der Exacte Wetenschappen
De Boelelaan 1081a
1081 HV Amsterdam
Coordinates of Miek Messerschmidt
Visiting address: Mathematical Institute, Leiden University, Niels Bohrweg 1 (Snellius building).
Email: miek@math.leidenuniv.nl
tel: 071 5277047
Postal address:
Mathematical Institute
Leiden University
P.O. Box 9512
2300 RA Leiden
The text we will use is the book by J.B. Conway: A course in Functional Analysis, second edition. Springer Verlag.
We assume some prior knowledge of Functional Analysis, although most of what we assume will be covered fairly quickly in the first three weeks. The goal of the course is to cover as much of the book as we can.
Examination
In principle exclusively on the basis of homework: an average grade of
at least 5.5 for the homework assignments implies passing. However,
although it is unlikely to occur, the lecturers reserve the right to
invite individual students for an oral exam if necessary and base the
final grade on both the homework and the oral exam.
Homework
The homework will be assigned at every even numbered
lecture, with a deadline three workable weeks later. It can be handed
in at the lecture, by email or by regular mail, the latter two
preferably directly addressed to the assistant. If it is handed in
between three weeks and four weeks later it is still accepted, but one
point will be deducted from the grade for that particular assignment.
Assignments will be accepted later than four weeks after the assignment
only in case of clear farce majeure. The corrected homework will be
handed back five weeks after the assignment, or sent back by regular
mail in case of the assignments 6 and 7 (please write your postal
address on the work).
Schedule and assignments:
Homework 1: assigned September 13, deadline October 4:
Homework 6: assigned November 29, deadline January 3. Note that there
is no "one-point-deduction-week" after these five weeks!!
To be handed in at a lecture, or with Blaz Mramor. Please write
your postal address on your work
Material actually covered
Lecture 1 (Andre Ran, September 6):
We covered Chapter 1 and Appendix A.2. Much of the material in Chapter 1 should be familiar from an introductory course, with the possible exception of working with a basis of a Hilbert space which is not necessarily countable. For such (non-separable) spaces it is relevant to give meaning to generalized series as in Lemma 4.12, and in order to do this we need the material on nets and their convergence in Appendix A.2. It is important to understand this material on nets, not just because we needed them for Hilbert spaces, but more generally since in non-metric spaces -- which occur naturally on many occasions -- sequences are no longer sufficient to work with, but nets are. As an example, Proposition A.2.4 is false when nets are replaced with sequences: it is always true that a continuous map transforms convergent sequences into convergent sequences, but this does not characterize continous maps in general.
Lecture 2 (André Ran, September 14, 2010):
This week we covered the largest part of Sections 1-3 from Chapter II, and the first three sections of Chapter III. Bounded linear operators were discussed on Hilbert spaces, with the important observation that boundedness and continuity are equivalent. Properties of the adjoint were discussed, and several classes of operators that are characterized by a certain relation between the operator and its adjoint were introduced (selfadjoint, normal, isometric, unitary). Projections and idempotents were introduced. A slight warning is in place: for Conway a projection is always an orthogonal projection. After this, Banach spaces were introduced. It is important that you try to understand all the examples given in the first section of chapter III. Linear operators were introduced, and finally, several propositions were discussed in which finite dimensionality plays a role somewhere.
Lecture 3 (André Ran, September 21, 2010):
In the third week we covered the most important theorems from the introductory courses: the Hahn-Banach theorem, the open mapping and closed graph theorems, the inverse mapping theorem and the uniform boundedness principle. All this is nicely explained in the book in sections 3.5, 3.6 and 3.10-14.
Lecture 4 (Marcel de Jeu, September 27, 201q):
We concentrated on IV.1 mostly, giving a thorough motivation of and
introduction to locally convex spaces. Among the things to remember are
how to introduce the topology in terms of a separating family of
seminorms, what it means for a net to converge in the resulting topology,
and Proposition IV.1.15 which validates the name "locally convex
space".
To get some practice in arguing in a TVS, we also proved the equivalence
of (b) and (d) in Theorem IV.3.1. Conway claims that his earlier proof for
normed spaces works here again, but if you look at III.5.3 you will see
that this is not so obvious. At that point he uses crucially that there is
only one norm topology on a finite dimensional space. Now it is, in fact,
also the case that there is only one topology which makes a finite
dimensional space into a TVS (so that, in particular, each finite
dimensional TVS is normable), but this is not proved in the book. So his
proof is not entirely satisfactory, and since the characterization of
continuous linear functionals as those linear functionals having a closed
kernel is important later on, we have given an independent proof.
We also covered example IV.3.16. This is a topological vector space having
only itself as a non-empty open convex subset, and consequently only 0 as
a continuous functional. As we will see, this is very far from the locally
convex case where continuous linear functionals exist in abundance. Our
geometric intuition lets us down in this example, where each element of
the space is a convex combination of elements of an arbitrarily small open
metric ball centered at the origin.
Next time we will finish the paragraphs 1-3 of Chapter IV and make a dent
in Chapter V.
Lecture 5 (Marcel de Jeu, October 4, 2011):
We finished Chapter IV.
In IV.2, Proposition 2.1 is the result most often used. Metrizable spaces
are good to work in. If a TVS is metrizable and then complete as a metric
spaces (these are the so-called Fréchet spaces), then the Baire
Category
Theorem holds and one can prove the consequences thereof as we already
know them
for Banach spaces, such as the Open Mapping Theorem, Closed Graph Theorem,
Bounded Inverse Theorem and Uniform Boundedness Principle. These are all
true for Fréchet spaces (but this is not proved in the book).
Section IV.3 starts with separation theorems for a TVS. Theorem 3.7 is
then about as good as one can get, and the key transition to separation
theorems for LCS is made in Theorem 3.9. Separation theorems imply the
usual Hahn-Banach theorems as we know them from the normed case: after
applying a separation theorem, one simply observes that a linear
functional which maps a linear subspace into a half plane (or half line)
in the field must actually be zero. This gives the basic HB-result 3.15
and (this is not in the book!) we used that one in turn to show that a
continuous linear functional on a (not necessarily closed) linear subspace
of a LCS extends to a continuous linear functional on the whole space. The
moral of all this is, that in locally convex spaces one has a dual which
is large enough so that the "usual" basic theorems about continuous linear
functionals hold: local convexity gives Hahn-Banach type results.
Incidentally, some authors call a Fréchet spaces a space which is metrizable and then
complete, as well as locally convex. It is now clear why such spaces are
very good spaces to work in: one has the consequences of the category theorem
at one's disposal, as well as a large dual to work with.
We made a start with Chapter V, by showing how a separating linear
subspace V of the algebraic dual of a vector space X can be used to
introduce a locally convex topology on X, such that the (continuous) dual
of X is precisely the V one started with. Special cases are the weak
topology on a LCS X and the weak*-topology on the dual X*. Theorem 1.2 and
1.3 simply reflect the general principle of the construction: the dual is
the V one started with.
We also showed that, if two locally convex topologies on X give the same
dual, then the closure of a convex subset of X will be the same in both
topologies. The items 1.4 and 1.5 follow from this general result. A
(deeper) theorem of Mackey (not in the book) states that two locally
convex topologies with the same dual also have the same bounded subsets
(no convexity assumption on these sets).
Next time we will move forward to, amongst others, the Banach-Alaoglu
theorem and the Krein-Milman theorem. These two combined are very a
powerful tool to show existence of objects of a desired type in many
situation.
Lecture 6 (Marcel de Jeu, October 11, 2011)
We concluded the mini series of three lectures on locally convex spaces by
covering more material from Chapter V and giving an application to ergodic
theory (which is neither in the book nor in he homework).
The first main result was the Bipolar Theorem 1.8, which together with 1.9
implies that the closed (=weakly closed) linear subspaces of a LCS X are
in bijection with the weak*-closed linear subspaces in X*. The bijection
is given by taking annihilators (these are the polars and prepolars of
linear subspaces).
The second main result (we skipped V.2) was the Banach-Alaoglu theorem.
We formulated the most general theorem, stating that the polar of an open
neighborhood
of 0 in a TVS X is always compact in the weak*-topology. This follows from
Tychonov's theorem: see 3.15 in Rudin's FA book for the details of the
proof.
In Conway the result is stated and proved for normed spaces as V.3.1, and
given as an exercise for locally convex spaces as exercise V.3.2, but it
is actually true for topological vector spaces in general.
The BA-theorem has consequences for reflexivity by relating it to weak
compactness: see 4.2. In this context the (deep) Eberlein-Smulian theorem
V.13.1
was mentioned, which states that for the weak topology on a normed space
the notions of compactness and sequential compactness of a subset of X are
the same. The same holds for metric spaces, but for general topological
spaces the notions are independent: there are spaces which are compact but
not sequentially compact, and there are spaces which are sequentially
compact, but not compact.
We noted the metrizability result 5.1 and mentioned that, for a Banach
space X, the metrizability of X in the weak topology and the metrizability
of X* in the weak*-topology are both equivalent to the space being finite
dimensional. This is not entirely trivial; see 2.5.14 and 2.6.12 in
Megginson's "An introduction to Banach space theory" for a proof.
The third main result was the Krein-Milman theorem: 7.4+7.8.
The Banach-Alaoglu theorem and the Krein-Milman theorem often occur in a
highly powerful combination to show that certain "desired" objects exist,
by first noting that these desired objects correspond to extremal points
of a non-empty convex set in the dual of a LCS, next noting
(Banach-Alaoglu) that this set is weak*-compact, and finally applying the
Krein-Milman theorem to conclude that there are extremal points as
needed.
We gave an example of such a concrete application in ergodic theory, by
showing that for any continous map from a compact Hausdorff space to
itself there is always an ergodic probability measure.
In two week's time André Ran will start a new mini series of four
lectures.
Lecture 7 (André Ran, October 25, 2011):
We started with Chapter VI, Sections 1 and 3. Section 1 on the adjoint of an operator acting between Banach spaces was covered up to and including Proposition 1.9. Next we introduced compact operators and completely continuous operators, and showed what the relation between the two concepts are (Proposition 3.3). We showed that if A is compact then its adjoint is compact. Also we showed that the set of compact operators is an ideal in the set of all bounded linear operators. Further, we discussed whether or not the closure of the set of operators with finite dimensional range is equal to the set of compact operators. Here we proved nothing, but it is pointed out that the discussion on pages 175 and 176 is very interesting.
The next topic was an introduction to Banach algebras, Sections VII.1 and 2. Examples were given, and we discussed the notions of ideals, quotients and maximal ideals. An important point is that the set of invertible elements in a Banach algebra is open.
Lecture 8 (André Ran, November 1, 2011):
This week the spectrum of an element in a Banach algebra was the main topic. We covered the Riesz functional calculus, the spectral mapping theorem, and we specialized to the Banach algebra of bounded linear operators on a Banach space. In particular, we showed that the boundary of the spectrum consists of approximate point spectrum.
Lecture 9 (André Ran, November 8, 2011):
We discussed the spectral properties of compact operators on Hilbert spaces. Contrary to the general Banach space situation, here we have that every compact operator is the limit in norm of a sequence of finite rank operators. We showed that every non-zero point in the approximate point spectrum of a compact operator T is actually an eigenvalue, and that for every non-zero eigenvalue the corresponding eigenspace is finite dimensional. Next, it was shown that if a non-zero complex number z is not an eigenvalue of T, and if the complex conjugate of z is not an eigenvalue of the adjoint of T, then z is not in the spectrum of T. It was then shown that the second condition is superfluous. Finally, the spectral theorem for compact selfadjoint operators was discussed.
Lecture 10 (André Ran, November 15, 2011):
The first highlight of this lecture was Section VII.8 on abelian Banach algebras, where we showed that every abelian Banach algebra A "is" a subalgebra of the set of continuous functions on a compact Hausdorff space X. The set X is actually the maximal ideal space, that is, the set of non-zero homomorphism from A to the complex numbers, equipped with the weak-star topology.
We then did a very brief introduction to Fredholm operators (chapter XI, only the basics of Sections 1,2,3). It was shown that Conway's definition of Fredholm operators is equivalent to the more usual one that states that A is Fredholm if and only if ker A is finite dimensional, ran A isclosed and ker A* is finite dimensional. We proved the logarithmic property of the Fredholm index. Finally we discussed two perturbation results, stating that the index is constant if one perturbs a Fredholm operator with either a compact or a small bounded operator.
Finally, we discussed Section II.6 on Sturm Liouville operators as an application of the spectral theorem of compact selfadjoint operators on a Hilbert space.
Lecture 11 (Marcel de Jeu, November 22, 2011):
We started working on C*-algebras and covered Sections 1 and 2 of Chapter
VIII. The highlights are the automatic continuity of *-homomorphisms
(Prop. 1.11 (d); this will be improved later on when we show that a
non-zero *-homomorphism has norm 1 and a C*-subalgebra as image),
Proposition 1.14 (Spectral permanence: the spectrum does not depend on the
algebra, except perhaps the complex number zero which in the non-unital
case may appear or disappear), the commutative Gelfand Naimark Theorem 2.1
(the commutative C*-algebras are, via the Gelfand transform, precisely the
algebras C_0(X) with X locally compact Hausdorff) and the continuous
functional calculus for normal elements (Theorem 2.6 and 2.7). The
definition of the continuous functional calculus looks a bit involved at
first sight, but if you keep the homeomorphism in Proposition 2.3 and the
commutative Gelfand Naimark theorem in mind, it is completely obvious that
the isomorphism which defines the continuous functional calculus does in
fact exist, and that the unital C*-subalgebra generated by a normal
element is isometrically *-isomorphic to the continuous functions on the
spectrum of that element.
Lecture 12 (Marcel de Jeu, November 29, 2011):
We continued our study of basic properties of C*-algebras in Sections 3, 4
and 5 of Chapter VIII. Most of these ultimately rely on the commutative
Gelfand-Naimark theorem (2.1) which provides, amongst others, a powerful
functional calculus.
In Section 3 we concentrated on proving Proposition 3.7: the positive
elements are a closed convex cone. This gives a partial ordering on the
self-adjoint elements of A, which then becomes a partially ordered real
vector space. Theorem 3.8 is important, because it shows that the two
notions of positivity in B(H) coincide. Although not without interest, we
did not cover the decomposition results 3.4 and 3.5 for reasons of time.
The highlights in Section 4 are 4.3 (closed two-sided ideals are
self-adjoint), 4.6 (the quotient norm has the C*-property) and 4.8 (the
image of a *-homomorphism is a C*-subalgebra). Although not mentioned
explicitly in the book, it then also follows immediately that A/Ker \rho
is isomorphic to \rho (A) as C*-algebras, i.e., the usual isomorphism
theorem holds in the strongest possible sense. In addition, it is also
true that a non-zero *-homomorphism has norm 1. These results are all
easily formulated and remembered, but they required quite some work to
prove!
We made a dent in Section 5, by defining cyclic representations and
observing that these are the building blocks of all representations
(Theorem 5.9). The subsequent step, in the next lecture, will be on the
remaining material in Section 5 (from 5.10 onward). We will see how there
is a natural bijection between positive functionals on A and unitary
equivalence classes of cyclic representations with a cyclic vector given
(GNS-theorem 5.14), and also that there are sufficiently many positive
functionals to build an injective (hence: isometric) representation of A
(5.17, this is the famous Gelfand-Naimark theorem which states that the
closed *-invariant subalgebras of B(H) are the only C*-algebras, up to
isomorphism). After that, we continue in Chapter IX.
Lecture 13 (Marcel de Jeu, December 6, 2011):
We covered Section 5 of Chapter VIII. The GNS-theorem 5.14, slightly
reformulated, shows that there is a bijection between positive functionals
and unitary equivalence classes of cyclic representations with a given
cyclic vector. So, to construct representations of a C*-algebra "all" one
needs are positive functionals. These can be characterised as the
continous functionals which assume their norm at the identity element
(this precise version was proved during the lecture). This
characterisation implies that any Hahn-Banach extension of a positive
functional from a C*-subalgebra to the whole algebra is automatically
positive, and since it is easy to find positive functionals on commutative
C*-subalgebras (these are the positive measures on the spectrum) one has a
rich supply of positive functionals, hence of representations. With all
this at one's disposal, the proof of the Gelfand-Naimark theorem 5.17 is
then easy: one rapidly concludes that there is an injective *-homomorphism
into some B(H), for some large Hilbert space H, and from the general
theory we then already know that this must be an isometric *-embedding.
Hence, after the fact, any C*-algebras "is" a C*-algebra of operators and
in retrospect this puts some of the earlier results, which were obvious
for operators, into perspective.
Lecture 14 (Marcel de Jeu, December 13, 2011):
We gave a detailed demonstration of the main results of Section 1 in
chapter IX, namely Proposition 1.12 and Theorem 1.14. Our proof was a bit
different from Conway's, as we avoided invoking Proposition V.4.1
(Goldstine's Theorem) in the proof of Theorem 1.14 and also the notion of
the weak operator topology. The key tools in both approaches, however,
remains the Riesz Representation theorem describing the dual of C(X),
which yields the complex valued measures in the proofs, and the well-known
bijection between the bounded sesquilinear forms on a Hilbert space and
the bounded operators on that space, which enables one to define operators
from the measures.
The moral of Proposition 1.12 and Theorem 1.14, taken together, is that
there is a natural bijection between representations of C(X) and regular
spectral measures on the Borel sets of the compact Hausdorff space X, and
that a representation of C(X) can be extended to a representation of the
bounded Borel functions on X (using Proposition 1.12 again).
After the work done in Section 1, the version (there are many) of the
spectral theorem which is given as Theorem 2.2 is an easy consequence;
this concluded our work on spectral theory. One of the ways to remember
the spectral theorem is that the continuous functional calculus can be
extended to a Borel functional calculus, i.e., to a representation of the
bounded Borel functions on the spectrum of N. The orthogonal projections
of the spectral measure are then simply the images of the characteristic
functions under this extended representation. It is then, e.g., obvious
that N can be approximated by linear combinations of mutually orthogonal
projections: simply approximate the function z uniformly on the spectrum
by step functions. Since the extended representation is contractive, the
images of the step functions (which are the linear combinations of
projections as required) will approximate the image of the function z
(i.e., approximate N) in the operator norm.
It should be obvious from these two sections that spectral theory and
measure theory are intimately connected. This becomes even more obvious in
the remaining sections in this chaper (which we did not cover),
culminating in the Theorems 10.20 and 10.21, where normal operators on a
separable Hilbert space are classified up to unitary equivalence.
Multiplication operators on an L_2-space are obvious examples of normal
operators and part of the statement of Theorems 10.20 and 10.21 is that
every normal operator on a Hilbert space is built up from building blocks
of this type.