## 414/514 Advanced Analysis (Analysis I) – Syllabus

August 20, 2014

Instructor: Andrés E. Caicedo
Fall 2014

Time: MWF 10:30-11:45 am.
Place: Mathematics building, Room 139.

Contact Information

• Office: 239-A Mathematics building.
• Phone number: (208)-426-1116. (Not very efficient.)
• Office Hours: W 12:00-1:15 pm. (Or by appointment.)
• Email: caicedo@math.boisestate.edu

Text
We will use three textbooks and complement with papers and handouts for topics not covered there.

• MR1886084 (2003e:00005).
Pugh, Charles Chapman
Real mathematical analysis.
Undergraduate Texts in Mathematics. Springer-Verlag, New York, 2002. xii+437 pp.
ISBN: 0-387-95297-7.
• MR0655599 (83j:26001).
van Rooij, A. C. M.; Schikhof, W. H.
A second course on real functions.
Cambridge University Press, Cambridge-New York, 1982. xiii+200 pp.
ISBN: 0-521-23944-3; 0-521-28361-2.
Gelbaum, Bernard R.; Olmsted, John M. H.
Counterexamples in analysis.
Corrected reprint of the second (1965) edition. Dover Publications, Inc., Mineola, NY, 2003. xxiv+195 pp.
ISBN: 0-486-42875-3.

The book by van Rooij and Schikhof will be our primary reference, supplemented naturally by the Counterexamples book.  The book assumes some knowledge beyond what is covered in our undergraduate course Math 314: Foundations of Analysis, and does not cover the theory in dimension $n>1$; for these topics, we will follow Pugh’s text.

Contents
Math 414/514 covers Analysis on Euclidean spaces (${\mathbb R}^n$) with emphasis on the theory in dimension one. The approach is theoretical, as opposed to the more computational approach of calculus, and a certain degree of mathematical maturity is required. The course is cross-listed, and accordingly the level will be aimed at beginning graduate students.

From the Course Description on the Department’s site:

Introduction to fundamental elements of analysis on Euclidean spaces including the basic differential and integral calculus. Topics include: infinite series, sequences and series of function, uniform convergence, theory of integration, implicit function theorem and applications.

Here is a short list of topics we expect to cover. This list may change based on students’ interest:

1. Set theoretic preliminaries.
• Cantor’s approach to infinite cardinalities. Countable vs. uncountable sets. Sets of size continuum. The Bernstein-Cantor-Schröder theorem.
• The axiom of choice. Zorn’s lemma. Countable and dependent choice.
• Transfinite recursion. The first uncountable ordinal $\omega_1$.
2. Axiomatization and construction of the set of reals.
• The least upper bound property; uniqueness of $\mathbb R$ up to isomorphism.
• Dedekind cuts, and complete orders.
• Metric spaces, and Cauchy completions. Banach contraction mapping theorem.
3. Topology on $\mathbb R$.
• Open and closed sets. Compact sets and Cantor sets. Baire space.
• Borel sets. Analytic sets.
• Notions of smallness.
• Meagerness and the Baire category theorem. The Baire-Cantor stationary principle.
• Sets of Jordan content zero and of measure zero.
• Introduction to the theory of strong measure zero sets.
4. Continuity.
• Sets of discontinuity of functions.
• Monotonicity. Functions of bounded variation.
5. Differentiability.
• The problem of characterizing derivatives. Baire class one functions. The intermediate value property. Sets of continuity of derivatives.
• The mean value theorem. L’Hôpital’s rule.
• The dynamics of Newton’s method.
• The Baire hierarchy of functions.
• Continuous nowhere differentiable functions.
6. Power series.
• Real analytic functions. Taylor series.
• $C^\infty$ functions. Zahorsky’s characterization of the sets of points where a $C^\infty$ function fails to be analytic.
7. Integration.
• Riemann integration. Lebesgue’s characterization of Riemann integrability.
• Weierstraß approximation theorem.
• Lebesgue integration. The fundamental theorem of calculus.
• The Henstock-Kurzweil integral. Denjoy’s approach to reconstructing primitives.
8. Introduction to multivariable calculus.
• (Frechet) derivatives.
• The inverse and implicit function theorems.

Based on homework. No late homework is allowed. Collaboration is encouraged, although students must turn in their own version of the solutions, and give credit to books/websites/… that were consulted and people with whom the problems where discussed.

There will be no exams. However, an important component of being proficient in mathematics is a certain amount of mental agility in recalling notions and basic arguments. I plan to assess these by requesting oral presentations of solutions to some of the homework problems throughout the term. If I find the students lacking here, it will be necessary to have an exam or two. The final exam is currently scheduled for Wednesday, December 17, 2014, 12:00-2:00 pm.

Additional information will be posted in this blog, and students are encouraged to use the comments feature. Please use full names, which will simplify my life filtering spam out.

## On strong measure zero sets

December 6, 2013

I meant to write a longer blog entry on strong measure zero sets (on the real line $\mathbb R$), but it is getting too long, so it may take me more than I expected. For now, let me record here an argument showing the following:

Theorem. If $X$ is a strong measure zero set and $F$ is a closed measure zero set, then $X+F$ has measure zero.

The argument is similar to the one in

Janusz Pawlikowski. A characterization of strong measure zero sets, Israel J. Math., 93 (1), (1996), 171-183. MR1380640 (97f:28003),

where the result is shown for strong measure zero subsets of $\{0,1\}^{\mathbb N}$. This is actually the easy direction of Pawlikowski’s result, showing that this condition actually characterizes strong measure zero sets, that is, if $X+F$ is measure zero for all closed measure zero sets $F$, then $X$ is strong measure zero. (Since this was intended for my analysis course, and I do not see how to prove Pawlikowski’s argument without some appeal to results in measure theory, I am only including here the easy direction.) Pawlikowski’s argument actually generalizes an earlier key result of Galvin, Mycielski, and Solovay, who proved that a set $X$ has strong measure zero iff it can be made disjoint from any given meager set by translation, that is, iff for any $G$ meager there is a real $r$ with $X+r$ disjoint from $G$.

I proceed with the (short) proof after the fold.

## Riemann on Riemann sums

November 16, 2013

Though Riemann sums had been considered earlier, at least in particular cases (for example, by Cauchy), the general version we consider today was introduced by Riemann, when studying problems related to trigonometric series, in his paper Ueber die Darstellbarkeit einer Function durch eine trigonometrische Reihe. This was his Habilitationsschrift, from 1854, published posthumously in 1868.

Riemann’s papers (in German) have been made available by the Electronic Library of Mathematics, see here. The text in question appears in section 4, Ueber den Begriff eines bestimmten Integrals und den Umfang seiner Gültigkeit. The translation below is as in

• A source book in classical analysis. Edited by Garrett Birkhoff. With the assistance of Uta Merzbach. Harvard University Press, Cambridge, Mass., 1973. MR0469612 (57 #9395).

Also zuerst: Was hat man unter $\displaystyle \int_a^b f(x) \, dx$ zu verstehen?

Um dieses festzusetzen, nehmen wir zwischen $a$ und $b$ der Grösse nach auf einander folgend, eine Reihe von Werthen $x_1, x_2,\ldots, x_{n-1}$ an und bezeichnen der Kürze wegen $x_1 - a$ durch $\delta_1$, $x_2 - x_1$ durch $\delta_2,\ldots,$ $b - x_{n-1}$ durch $\delta_n$ und durch $\varepsilon$ einen positiven ächten Bruch.  Es wird alsdann der Werth der Summe

$\displaystyle S = \delta_1 f(a + \varepsilon_1 \delta_1) + \delta_2 f(x_1 + \varepsilon_2 \delta_2) + \delta_3 f(x_2 + \varepsilon_3 \delta_3) +\cdots$ $\displaystyle +\delta_n f(x_{n-1} +\varepsilon_n \delta_n)$

von der Wahl der Intervalle $\delta$ und der Grössen $\varepsilon$ abhängen.  Hat sie nun die Eigenschaft, wie auch $\delta$ und $\varepsilon$ gewählt werden mögen, sich einer festen Grenze $A$ unendlich zu nähern, sobald sämmtliche $\delta$ unendlich klein werden, so heisst dieser Werth $\displaystyle \int_a^b f(x) \, dx$.

In Birkhoff’s book:

First of all: What is to be understood by $\displaystyle \int_a^b f(x)\,dx$?

In order to establish this, we take the sequence of values $x_1,x_2,\ldots, x_{n-1}$ lying between $a$ and $b$ and ordered by size, and, for brevity, denote $x_1 - a$ by $\delta_1$, $x_2 - x_1$ by $\delta_2,\ldots,$ $b - x_{n-1}$ by $\delta_n$, and proper positive fractions by $\varepsilon_i$. Then the value of the sum

$\displaystyle S = \delta_1 f(a + \varepsilon_1 \delta_1) + \delta_2 f(x_1 + \varepsilon_2 \delta_2) + \delta_3 f(x_2 + \varepsilon_3 \delta_3) +\cdots$ $\displaystyle +\delta_n f(x_{n-1} +\varepsilon_n \delta_n)$

will depend on the choice of the intervals $\delta_i$ and the quantities $\varepsilon_i$. If it has the property that, however the $\delta_i$ and the $\varepsilon_i$ may be chosen, it tends to a fixed limit $A$ as soon as all the $\delta_i$ become infinitely small, then this value is called $\displaystyle \int_a^b f(x) \, dx$.

(Of  course, in modern presentations, we use $\Delta_i$ instead of $\delta_i$, and say that the $\delta_i$ approach $0$ rather than become infinitely small. In fact, we tend to call the collection of data $x_1,\dots,x_{n-1}$, $\varepsilon_1,\dots,\varepsilon_n$ a tagged partition of ${}[a,b]$, and call the maximum of the $x_{i+1}-x_i$ the mesh or norm of the partition.)

## Analysis – HW 5 – Newton’s method

November 16, 2013

This set is due Friday, December 6, at the beginning of lecture.

Newton’s method was introduced by Newton on De analysi in 1669. It was originally restricted to polynomials; his example in Methodus fluxionum was the cubic equation

$x^3-2x-5=0.$

Raphson simplified its description in 1690. The modern presentation, in full generality, is due to Simpson in 1740. Here, we are mostly interested in the dynamics of Newton’s method on polynomials.

## Weierstrass function

November 7, 2013

Weierstrass function from 1872 is the function $f=f_{a,b}$ defined by

$\displaystyle f(x)=\sum_{n=0}^\infty a^n\cos(b^n\pi x)$.

Weierstrass showed that if

• $0,
• $b$ is an odd positive integer, and
• $\displaystyle ab>1+\frac32\pi$,

then $f$ is a continuous nowhere differentiable function. Hardy proved in 1916 that one can relax the conditions on $a,b$ to

• $0,
• $b>1$, and
• $ab\ge 1$.

Here, I just want to show some graphs, hopefully providing some intuition to help understand why we expect $f$ to be non-differentiable. The idea is that the cosine terms ensure that the partial sums  $\displaystyle f(m,x)=\sum_{n=0}^m a^n\cos(b^n\pi x)$, though smooth, have more and more “turns” on each interval as $m$ increases, so that in the limit, $f$ has “peaks” everywhere. Below is an animation (produced using Sage) comparing the graphs of $f(m,x)$ for $0\le m<20$ (and $-10\le x\le 10$), for $a=1/2$ and $b=11$, showing how the bends accumulate. (If the animations are not running, clicking on them solves the problem. As far as I can see, they do not work on mobiles.)

Below the fold, we show the same animation, zoomed in around $0$ by factors of $100$, $10^4$, and $10^6$, respectively, illustrating the fractal nature of $f$.

## Continuous nowhere differentiable functions

November 7, 2013

Following a theme from two years ago, we will have a final project for this course, due Wednesday, December 18, by noon, but feel free (and encouraged) to turn it in earlier. (As discussed in lecture, the project is voluntary for some of you. Contact me if you are not sure whether it is required or voluntary for you.)

There are many excellent sources on the topic of continuous nowhere differentiable functions. Johan Thim’s Master thesis, written under the supervision of Lech Maligranda, is available online, here, but feel free to use any other sources you find relevant.

Please choose an example of a continuous nowhere differentiable function, either from Thim’s thesis or elsewhere, and write (better yet, type) a note on who it is due to and what the function is, together with complete proofs of continuity and nowhere differentiability. Though not required, feel free (and encouraged) to add additional information you consider relevant for context.

(For an example of what I mean by relevant additional information: Weierstrass function is $\displaystyle f(x)=\sum_{n=0}^\infty a^n\cos(b^n\pi x)$ where $0, $b$ is an odd positive integer, and $\displaystyle ab>1+\frac32\pi$. It may be interesting to add a discussion of precisely what conditions are needed from $a,b$ to ensure (continuity and) nowhere differentiability; Weierstrass original requirements are more restrictive than necessary. For another example, Schoenberg functions, discussed in Thim’s thesis, give a natural example of a space filling curve, so consider including a proof of this fact.)

Please take this project very seriously (in particular, do not copy details from books or papers, I want to see your own version of the details as you work through the arguments). Feel free to ask for feedback as you work on it; in fact, asking for feedback is a good idea. Do not wait until the last minute. At the end, it would be nice to make at least some of the notes available online, please let me know when you turn it in whether you grant me permission to host your note on this blog.

Here is a list of the projects I posted on the blog, from last time:

Contact me (by email) as soon as you have chosen the example you will work on, to avoid repetitions; I will add your name and the chosen example to the list below as I hear from you.

List of projects:

• Joe Busick: Katsuura function.
• Paul Carnig: Darboux function.
• Joshua Meier: A variant of Koch’s snowflake.
• Paul Plummer: Lynch function.
• Veronica Schmidt: McCarthy function.

## Credit

November 5, 2013

I recognize I owe much to Messrs. Bernoulli’s insights, above all to the young, currently a professor in Groningue. I did unceremoniously use their discoveries, as well as those of Mr. Leibniz. For this reason I consent that they claim as much credit as they please, and will content myself with what they will agree to leave me.

L’Hôpital, in the preface (page xiv) of his Analyse des Infiniment Petits pour l’Intelligence des Lignes Courbes (1696), the first calculus textbook, published anonymously. (A posthumous second edition, from 1716, identifies L’Hôpital as the author.)