Category Archives: Advanced Analysis, 20125401

Advanced Analysis, announcement

Advanced Analysis, 20125401

I have been writing my own lecture notes (based, of course, on an array of well known references and also on some of my notes from my graduate studies) for the course Advanced Analysis. I have decided to do this because, on the one hand, there was no particular text that I wanted to follow, while on the other hand I wanted my students to have a convenient reference for the material in the course. I hope these notes were instructive to some readers of the blog.

During the second half of the course I will follows Arveson’s book “A Short Course on Spectral Theory”, besides some applications and/or examples that I will want to occasionally throw in. So I will post my notes on much rarer occasions.

Advanced Analysis, Notes 11: Banach spaces (weak topologies, Alaoglu’s theorem)

Advanced Analysis, 20125401, Functional analysis

Let $latex X$ be the Banach space $latex C([0,1])$ of continuous functions on the interval $latex [0,1]$ with the sup norm. Consider the following sequence of functions $latex {f_n}$ defind as follows. $latex f(0) = 0 $ and $latex f_n(1/(n+1)) = 1$ for all $latex n = 1, 2, ldots$,  $latex f_n$ is equal to zero on the interval between $latex 2/(n+1)$ and $latex 1$, and $latex f_n$ is linear in the intervals where we haven’t defined it yet (visualize!). The sequence is tending to zero pointwise, but the norm of $latex X$ does not detect this. The sequence tends to $latex 0$ in the $latex L^1$ norm, but the $latex L^1$ norm is not in the game. Can the Banach space structure of $latex X$ detect the convergence of $latex f_n$ to $latex 0$? Continue reading Advanced Analysis, Notes 11: Banach spaces (weak topologies, Alaoglu’s theorem) →

Advanced Analysis, Notes 10: Banach spaces (application: divergence of Fourier series)

Advanced Analysis, 20125401, Functional analysis,

Recall Theorem 6 from Notes 3:

Theorem 6: For every $latex f in C_{per}([0,1]) cap C^1([0,1])$, the Fourier series of $latex f$ converges uniformly to $latex f$. 

It is natural to ask how much can we weaken the assumptions of the theorem and still have uniform convergence, or how much can we weaken and still have pointwise convergence. Does the Fourier series of a continuous (and periodic) function always converge? In this post we will use the principle of uniform boundedness to see that the answer to this question is a very big NO.

Once again, we begin with some analytical preparations.  Continue reading Advanced Analysis, Notes 10: Banach spaces (application: divergence of Fourier series) →

Advanced Analysis, Notes 9: Banach spaces (the three big theorems)

Advanced Analysis, 20125401, Functional analysis

Until now we had not yet seen a theorem about Banach spaces — the Hahn–Banach theorems did not require the space to be complete. In this post we learn the three big theorems about operators on Banach spaces: the principle of uniform boundedness, the open mapping theorem, and the closed graph theorem. It is common that these three theorems are presented in texts on functional analysis under the heading “consequences of the Baire category theorem“.  Continue reading Advanced Analysis, Notes 9: Banach spaces (the three big theorems) →

Advanced Analysis, Notes 8: Banach spaces (application: weak solutions to PDEs)

Advanced Analysis, 20125401, Functional analysis, Partial differential equations, , ,

Today I will show you an application of the Hahn-Banach Theorem to partial differential equations (PDEs). I learned this application in a seminar in functional analysis, run by Haim Brezis, that I was fortunate to attend in the spring of 2008 at the Technion.

As often happens with serious applications of functional analysis, there is some preparatory material to go over, namely, weak solutions to PDEs.

Continue reading Advanced Analysis, Notes 8: Banach spaces (application: weak solutions to PDEs) →

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

Advanced Analysis, 20125401, Functional analysis, ,

Today we continue our treatment of the dual space $latex X^*$ of a normed space (usually Banach) $latex X$. We start by considering a wide class of Banach spaces and their duals.  Continue reading Advanced Analysis, Notes 7: Banach spaces (dual spaces and duality, Lp spaces, the double dual, quotient spaces) →

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

Advanced Analysis, 20125401, Functional analysis, , ,

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

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

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:

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

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 $latex C(X)$ or the operator norm on $latex B(H)$, 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.

Continue reading Advanced Analysis, Notes 6: Banach spaces (basics, the Hahn-Banach Theorems) →

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

Advanced Analysis, 20125401, Functional analysis, ,

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 $latex |AB| leq |A| |B|$.

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

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

Advanced Analysis, 20125401, Functional analysis, ,

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) →

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.

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