Noncommutative Analysis

Tag: Hilbert space

A First Course in Functional Analysis (my book)

She’hechiyanu Ve’kiyemanu!

My book, A First Course in Functional Analysis, to be published with Chapman and Hall/CRC, will soon be out. There is already a cover, check it out on the CRC Press website.

This book is written to accompany an undergraduate course in functional analysis, where the course I had in mind is precisely the course that we give here at the Technion, with the same constraints. Constraint number 1: a course in measure theory is not mandatory in our undergraduate program. So how can one seriously teach functional analysis with significant applications? Well, one can, and I hope that this book proves that one can. I already wrote before, measure theory is not a must. Of course anyone going for a graduate degree in math should study measure theory (and get an A), but I’d like the students to be able to study functional analysis before that (so that they can do a masters degree in operator theory with me).

I believe that the readers will find many other original organizational contributions to the presentation of functional analysis in this book, but I leave them for you to discover. Instructors can request an e-copy for inspection (in the link to the publisher website above), friends and direct students can get a copy from me, and I hope that the rest of the world will recommend this book to their library (or wait for the libgen version).

Advanced Analysis, Notes 5: Hilbert spaces (application: Von Neumann’s mean ergodic theorem)

In this lecture we give an application of elementary operators-on-Hilbert-space theory, by proving von Neumann’s mean ergodic theorem. See also this treatment by Terry Tao on his blog.

For today’s lecture we will require the following simple fact which I forgot to mention in the previous one.

Exercise A: Let A, B \in B(H). Then \|AB\| \leq \|A\| \|B\|.

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Advanced Analysis, Notes 4: Hilbert spaces (bounded operators, Riesz Theorem, adjoint)

Up to this point we studied Hilbert spaces as they sat there and did nothing. But the central subject in the study of Hilbert spaces is the theory of the operators that act on them. Paul Halmos, in his classic paper “Ten Problem in Hilbert Space“, wrote:

Nobody, except topologists, is interested in problems about Hilbert space; the people who work in Hilbert space are interested in problems about operators.

Of course, Halmos was exaggerating; topologists don’t really care much for Hilbert spaces for their own sake, and functional analysts have much more to say about the structure theory of Hilbert space then what we have learned. Nevertheless, this quote is very close to the truth. We proceed to study operators.  Read the rest of this entry »

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

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

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

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

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

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

(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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