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
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
- 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
- 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.)
Notes for colloquia given at BGSU to general mathematical audiences
(roughly equal numbers of faculty and graduate students). Lightweight
and not very technical.
- 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
- Brief Solutions to the Goblet Problems (8K)
- Solutions to a problem set used to accompany the "Reflect Divine" talk.
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.
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.
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
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.
- A few of the AMS-Fonts
(Blackboard Bold plus a couple of special symbols) are used
throughout; pretty but not essential.
- You will have to uncompress, unzip, or unstuff the
Short Course notes. Everything you're likely to need
can be found in the /tools directory of the
CTAN TeX Archives.
Neal L. Carothers