Category Archives: Functional analysis

Advanced Analysis, Notes 3: Hilbert spaces (application: Fourier series)

Advanced Analysis, 20125401, Functional analysis,

Consider the cube $latex K := [0,1]^k subset mathbb{R}^k$. Let $latex f$ be a function defined on $latex K$.  For every $latex n in mathbb{Z}^k$, the $latex n$th Fourier coefficient of $latex f$ is defined to be

$latex hat{f}(n) = int_{K} f(x) e^{-2 pi i n cdot x} dx ,$

where for $latex n = (n_1, ldots, n_k)$ and $latex x = (x_1, ldots, x_k) in K$ we denote $latex n cdot x = n_1 x_1 + ldots n_k x_k$.  The sum

$latex sum_{n in mathbb{Z}^k} hat{f}(n) e^{2 pi i n cdot x} $

is called the Fourier series of $latex f$. The basic problem in Fourier analysis is whether one can reconstruct $latex f$ from its Fourier coefficients, and in particular, under what conditions, and in what sense, does the Fourier series of $latex f$ converge to $latex f$.

One week into the course, we are ready to start applying the structure theory of Hilbert spaces that we developed in the previous two lectures, together with the Stone-Weierstrass Theorem we proved in the introduction, to obtain easily some results in Fourier series.

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Advanced Analysis, Notes 2: Hilbert spaces (orthogonality, projection, orthonormal bases)

Advanced Analysis, 20125401, Functional analysis

(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.

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Advanced Analysis, Notes 1: Hilbert spaces (basics)

Advanced Analysis, 20125401, Functional analysis

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.

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William Arveson

Arveson, Expository, Functional analysis, Operator theory,

William B. Arveson was born in 1934 and died last year on November 15, 2011. He was my mathematical hero; his written mathematics has influenced me more than anybody else’s. Of course, he has been much more than just my hero, his work has had deep and wide influence on the entire operator theory and operator algebras communities. Let me quickly give an example that everyone can appreciate: Arveson proved what may be considered as the “Hahn-Banach Theorem” appropriate for operator algebras. He did much more than that, and I will expand below on some of his early contributions, but I want to say something before that on what he was to me.

When I was a PhD student I worked in noncommutative dynamics. Briefly, this is the study of actions of (one-parameter) semigroups of *-endomorphisms on von Neumann algebras (in short E-semigroups). The definitive book on this subject is Arveson’s monograph “Noncommutative Dynamics and E-Semigroups”. So, naturally, I would carry this book around with me, and I would read it forwards and backwards. The wonderful thing about this book was that it made me feel as if all my dreams have come true! I mean my dreams about mathematics: as a graduate student you dream of working on something grand, something important, something beautiful, something elegant, brilliant and deep. You want your problem to be a focal point where different ideas, different fields, different techniques, in short, all things, meet.

When reading Arveson there was no doubt in my heart that, e.g., the problem classifying E-semigroups of type I was a grand problem. And I was blown away by the fact that the solution was so beautiful. He introduced product systems with such elegance and completeness that one would think that this subject has been studied for the last 50 years. These product systems were measurable bundles of operator spaces – which turn out to be Hilbert spaces! – that have a group like structure with respect to tensor multiplication. And they turn out to be complete invariants of E-semigroups on $latex B(H)$. The theory set down used ideas and techniques from Hilbert space theory, operator space theory, C*-algebras, group representation theory, measure theory, functional equations, and many new ideas – what more could you ask for? Well, you could ask that the new theory also contribute to the solution of the original problem.

It turned out that the introduction of product systems immensely advanced the understanding of E-semigroups, and in particular it led to the full classification of type I ones.

So Arveson became my hero because he has made my dreams come true. And more than once: when reading another book by him, or one of his great papers, I always had a very strong feeling: this is what I want to do. And when I felt that I gave a certain problem all I thought I had in me, and decided to move on to a new problem, it happened that he was waiting for me there too.

I wish to bring here below a little piece that I wrote after he passed away, which explains from my point of view what was one of his greatest ideas.

For a (by far) more authoritative and complete review of Arveson’s contributions, see the two recent surveys by Davidson (link) and Izumi (link).

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Functional Analysis – Introduction. Part II

Advanced Analysis, 20125401, Functional analysis, Operator theory,

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.

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Functional Analysis – Introduction. Part I

Advanced Analysis, 20125401, Expository, Functional analysis, Operator theory,

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).

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