### Advanced Analysis, Notes 7: Banach spaces (dual spaces and duality, Lp spaces, the double dual, quotient spaces)

Today we continue our treatment of the dual space $X^*$ of a normed space (usually Banach) $X$. We start by considering a wide class of Banach spaces and their duals.  Read the rest of this entry »

### Advanced Analysis, Notes 6: Banach spaces (basics, the Hahn-Banach Theorems)

Recall that a norm on a (real or complex) vector space $X$ is a function $\| \cdot \| : X \rightarrow [0, \infty)$ that satisfies for all $x,y \in X$ and all scalars $a$ the following:

1. $\|x\| = 0 \Leftrightarrow x = 0$.
2. $\|ax\| = |a| \|x\|$.
3. $\|x + y \| \leq \|x\| + \|y\|$.

A vector space with a norm on it is said to be a normed space. Inner product spaces are normed spaces. However, many norms of interest are not induced by an inner product. In fact:

Exercise A: A norm is induced by an inner product if and only if it satisfies the parallelogram law:

$\|x+y\|^2 + \|x-y\|^2 = 2 \|x\|^2 + 2\|y\|^2 .$

Instead of solving this exercise, you might prefer to read this old paper where Jordan and von Neumann prove this.

Using Exercise A, it is not hard to show that some very frequently occurring norms, such as the sup norm on $C(X)$ or the operator norm on $B(H)$, are not induced by inner products. The latter example shows that even if one is working in the setting of Hilbert spaces one is led to study other normed spaces. We now begin our study of normed spaces and, particular, Banach spaces.

### 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 $\|AB\| \leq \|A\| \|B\|$.

### 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 $n$th 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.