(Quick announcement: all lectures will from now on take place in room 201).
In the previous lecture, we learned the very basics of Hilbert space theory. In this lecture we shall go one little bit further, and prove the basic structure theorems for Hilbert spaces.
In this lecture and the next few lectures we will study the basic theory of Hilbert spaces. Hilbert spaces are usually studied over $latex mathbb{R}$ or over $latex mathbb{C}$. In this course, whenever we consider Hilbert spaces, we shall consider only complex Hilbert spaces, that is, spaces over $latex mathbb{C}$. The are two reasons for this. First, the results in this post hold equally well for real Hilbert spaces with similar proofs. Second, in some topics that we will discuss later the nice results only hold for complex spaces. So we will ignore real Hilbert spaces because they are essentially the same and also because they are fundamentally different!
Remark: The only situation I know where it is really important to concentrate on real Hilbert spaces when doing convex analysis (there must be others that I don’t know of). On the other hand, it is often convenient – indeed, we already did so in this course – to study real Banach spaces.
Continue reading Advanced Analysis, Notes 1: Hilbert spaces (basics)
In a previous post we discussed some of the history of functional analysis and we also said some vague things about its role in mathematics. In this second part of the introduction we will see an example of the spirit of functional analysis in action, by taking a close look at the Stone-Weierstrass approximation theorem.
Continue reading Functional Analysis – Introduction. Part II
I begin by making clear a certain point. Functional analysis is an enormous branch of mathematics, so big that it does not seem appropriate to call it “a branch”, it sometimes looks more like another tree. When I will talk below about functional analysis, I will mean “textbook functional analysis” and not “research functional analysis”. By this I mean that I will only refer to the core of the theory which is several decades old and which is more-or-less agreed to be the essential and basic part of the subject.
The goal of this post is to serve as an introduction to the course “Advanced Analysis, 201.2.5401”, which is a basic graduate course on (textbook) functional analysis. In the lectures I will only have time to give a limited description of the roots of the subject and the motivation will have to be brief. Here I will aim to describe what was the climate in which this tree grew, where are its roots and what are its fruits.
To prepare this introduction I am relying on the following sources. First and foremost, my love of the subject and my point of view on it were strongly shaped by my teachers, and in particular by Boris Paneah (my Master’s thesis advisor) and Baruch Solel (my PhD. thesis advisor). Second, I learned a lot on the subject from the book “Mathematical Thought from Ancient to Modern Times” by M. Kline and from the notes sections of Rudin’s and Reed-Simon’s books “Functional Analysis”.
And a warning to the kids: this is a blog, not a book, and if you really want to learn something go read the books (the books I mentioned have precise references).
I am planning to write a post that will be an introduction to the course “Advanced Analysis”, which I shall be teaching in the fall term. The introduction is to comprise two main themes: motivation and history. I was a little surprised to find out – as I was preparing the introduction – that, looking from the eyes of a student, the history of subject provided little motivation. I also began to oscillate between two opposite (and equally silly) viewpoints. The first viewpoint is that functional analysis is a big and respectable field of mathematics, which needs no introduction; let us start with the subject matter immediately since there is so much to learn. The second viewpoint is that there is absolutely no point in studying (or teaching) a mathematical theory without understanding its context and roots, or without knowing how it applies to problems outside of the theory’s borders. Pondering these, I found that I had some things to say before the introduction, which may justify the introduction or give it the place I intend.
Continue reading Functional analysis – a preface to the introduction
In the first term of the 2012/2013, I will be giving the course “Advanced Analysis” here at BGU. This is the department’s core functional analysis course for graduate students, though ambitious undergraduate students are also encouraged to take this course, and some of them indeed do. The price to pay is that we do not assume that the students know any functional analysis, and the only formal requisites are a course in complex variables and a course in (point set) topology, as well as a course in measure theory which can be taken concurrently. The price to pay for having no requisites in functional analysis, while still aiming at graduate level course, is that the course is huge: we have five hours of lectures a week. In practice we will actually have six hours of lectures a week, because I will go abroad in the middle of the semester to this conference and workshop in Bangalore. The official syllabus of the course is as follows:
Banach spaces and Hilbert spaces. Basic properties of Hilbert spaces. Topological vector spaces. Banach-Steinhaus theorem; open mapping theorem and closed graph theorem. Hahn-Banach theorem. Duality. Measures on locally compact spaces; the dual of C(X). Weak and weak-* topologies; Banach-Alaoglu theorem. Convexity and the Krein-Milman theorem. The Stone-Weierstrass theorem. Banach algebras. Spectrum of a Banach algebra element. Gelfand theory of commutative Banach algebras. The spectral theorem for normal operators (in the continuous functional calculus form).
I plan to cover all these topics (with all that is implicitly implied), but I will probably give the whole course a little bend towards my own area of expertise, especially in the exercises and examples. We do have to wait and see who the students are and what their background is before deciding precisely how to proceed. Some notes for the course will appear (in the English language) on this blog. The official course webpage (which is in Hebrew) is behind this link.