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 $latex A, B in B(H)$. Then $latex |AB| leq |A| |B|$.
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. Continue reading Advanced Analysis, Notes 4: Hilbert spaces (bounded operators, Riesz Theorem, adjoint)
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.
Continue reading Advanced Analysis, Notes 3: Hilbert spaces (application: Fourier series)
(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)