Lecture Notes:
Disclaimer and Description


These are classroom notes, not original research; they are offered here primarily as labor-saving tools for students and their teachers.

Much of this material has been culled from various articles, books, seminar notes, and classroom notes that I have accumulated over the years. Thus, portions have been taken from copyrighted material.

On the other hand, certain portions of these notes are original and are taken from my notes for an eventual book.

In any case, it would be very uncool to publish, or to use for profit, any or all of these notes. At the very least, I would have a conniption and never speak to you again.

You are otherwise welcome to use these notes in any way you see fit: Give copies to friends and family, "cut and paste" to save time typing up your own versions, or whatever. If you use the notes in seminars or courses of your own, I would greatly appreciate hearing your reactions.

See the bottom of this page for a description of what you need to process these notes.

Undergraduate Talks

Fixed Points and Iteration (7K)
An outline of a short talk delivered to an Honors class consisting of sophomore and junior math majors. Includes a few exercises and a few references.
Polygonal Functions (19K)
Notes for a short talk outlining Lebesgue's proof of the Weierstrass theorems using polygonal functions. Includes a number of exercises and references. Delivered at the Conference on Classical Analysis and Topology in the Undergraduate Curriculum, Miami University, Oxford, Ohio, September 30, 1994.

Colloquiua and Invited Talks

Notes for plenary addresses delivered at the Conference on Analysis in the Undergraduate Curriculum, Miami University, Oxford, Ohio, September 24--25, 2010. Lightweight and not very technical. (The talks are graphics-heavy, so I've posted them in pdf format.)

Think Deeply About Simple Things (2.4M)
A plea for more thorough coverage of the basics in calculus and beginning analysis (with references).
To Digress is Human, To Reflect Divine (3.7M)
Some strategies for promoting reflection in calculus and analysis courses (with references).
Brief Solutions to the Goblet Problems (8K)
Solutions to a problem set used to accompany the "Reflect Divine" talk.
Notes for colloquia given at BGSU to general mathematical audiences (roughly equal numbers of faculty and graduate students). Lightweight and not very technical.

Calculus at the Turn of the Century: The Age of Crisis (12K)
Brief outline; no references.
A Brief History of Functional Analysis (42K)
More detailed notes with references.

Graduate Seminars

Our local Analysis Seminar is attended by both faculty and graduate students (third year or later), representing a wide variety of backgrounds and interests. Each talk is meant to be informal, but more or less self-contained; each includes a short list of references.

Isometries on Lp (21K)
A discussion of the proof of the Banach-Lamperti theorem, characterizing the (linear, into) isometries on Lp.
Isometries on C(K) (23K)
Banach's proof of the Banach-Stone theorem, characterizing the (linear, onto) isometries of C(K), along with an outline of Stone's proof.
Compact and Weakly Compact Operators (24K)
A brief summary of facts concerning compact and weakly compact operators, culminating in a proof of the Davis-Figiel-Johnson-Pelczynski factorization theorem. Includes a few exercises.
The Spaces Lp and Hp for 0<p<1 (18K)
A brief summary of facts concerning the often overlooked range 0<p<1; particular emphasis on the failure of the Hahn-Banach theorem.

Short Courses

Each summer here at BGSU we offer a variety of survey and introductory courses to our graduate students on topics not usually covered in our mainstream courses. The audience is typically heterogeneous with regard to background and interest, and so it's important that each course be essentially self-contained.

These are often abbreviated or incomplete courses; more than enough material for a one quarter course, but not always enough for a full semester course. Nevertheless, the system seems to work. The non-specialist students are awarded extra breadth, while the specialists are granted a "leg up" on their reading.

A Short Course on Approximation Theory
Updated, expanded, largely self-contained version. Still somewhat brief but many more exercises.
An Introduction to Inequalities
My final course at BGSU; a brief introduction to classical inequalities. Largely self-contained, with a modest selection of exercises.

What You Need

Most of these notes are written in Plain TeX, but make use of a few extra goodies (all of which are available in the CTAN TeX Archives). A few of the more recent files were processed using Pdftex.

Neal L. Carothers