GENERAL INTRODUCTORY TEXTS

Mathematical Skills

For several years I was involved in the teaching of the module G1AMSK Mathematical Skills at the University of Nottingham, along with Dr J.A. Anderson. Dr Anderson covered mathematical language and proof, while I lectured methods of calculus (differentiation, integration and optimization). For Dr Anderson's part of the module the following books were the most useful.

Numbers and Proofs by R. Allenby;
Numbers, Sequences and Series by K. Hirst.

For my part of the module, the best book was the excellent Thomas' Calculus by George B. Thomas, Ross L. Finney, Maurice D. Weir and Frank Giordano.

There is an excellent web resource supporting this book.

Also good are the Schaum series books

Schaum's 3000 Solved Problems in Calculus by E. Mendelson;
Schaum's Outline of Calculus by F. Ayres and E. Mendelson.

REAL ANALYSIS

I have an associated web page at the University, G12RAN Real Analysis

An excellent text for most of the module as I taught it is Fundamentals of Mathematical Analysis, 2nd edition by R. Haggarty. However this book does not cover countability of sets.
Another book that gives a good overview of the material taught in my module is is the book Analysis by E. Kopp. For more detail you need to consult other texts, or perhaps the lecture notes of Professor Langley, available from the web page above.

LINEAR MATHEMATICS AND VECTOR SPACES

• G11LMA Linear Mathematics (part of the Core of the first year honours mathematics programme at the University of Nottingham)

  There are many good books in this area

  A highly recommended text, almost ideal for the whole module, is H. Anton and C. Rorres, Elementary Linear Algebra, applications version
  An excellent text for the second half of the module is the book in the Modular Mathematics series Matrices and Quadratic Forms by J. Bowers
  Another very useful text is W. Keith Nicholson's book, Elementary Linear Algebra.

• G12VSP Vector Spaces

  The most appropriate book for this module as I taught it is the book in the Modular Mathematics series Matrices and Quadratic Forms by J. Bowers

  It is also worth looking at two different books of Lang:
  • Introduction to Linear Algebra by S. Lang
  • Linear Algebra by S. Lang.

METRIC AND TOPOLOGICAL SPACES

Another subject that I have taught many times. I have an associated web page at the University, G13MTS Metric and Topological Spaces

The most appropriate book for this module as I have given it is Introduction to Metric and Topological Space by W. A. Sutherland. This book gives a very clear and accurate account of the material, and includes many illustrative examples.
Another excellent book is Bert Mendelson's Introduction to Topology

MEASURE AND INTEGRATION

I have taught this module several times at Nottingham. See my web page G1CMIN: Measure and Integration.

There is some overlap between this subject and Functional Analysis (see below). Certainly many introductory texts are appropriate for both.

My approach to integration is that used in the book Real and Complex Analysis by Walter Rudin. In fact I recommend all of Rudin's books. His proofs are very elegant and concise (the reader sometimes has to do a little work to check the less obvious steps in each proof but it helps to have such a trustworthy author!).

Functional Analysis

• Joel Feinstein's Research
• Books on Functional Analysis at Amazon.co.uk

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There are many fine books on Functional Analysis and Banach Algebras.

A good introductory text for Functional Analysis is Elements of the Theory of Functions and Functional Analysis by A. N. Kolmogorov and S. V. Fomin

I highly recommend the books of Walter Rudin, especially

Real and Complex Analysis,	Functional Analysis and	Fourier Analysis on Groups

Highly recommended is H. Garth Dales's book Banach Algebras and Automatic Continuity Garth Dales was my research supervisor and much of my own knowledge comes from reading early drafts of this excellent book!
If your library does not have a copy, I also recommend that you order Banach Algebras and the General Theory of *-algebras Vol 1 by Theodore W. Palmer. This book is an excellent reference book for the theory of Banach Algebras. If you want to know what the standard results are in a particular area, the chances are that there will be a section on it in Palmer's book. When you read through parts of this book you may well find out that there are some striking results in each area that you were unaware of before. ( Volume 2 of Palmer's book has now also been published.)
Some of the deepest and most useful results in Banach Algebras depend on the theory of Several Complex Variables. If you want to find out more about this, I recommend the book Several Complex Variables and Banach Algebras by H. Alexander and J. Wermer	.
Another classic text on Banach Algebras is the book of C.E. Rickart, General Theory of Banach Algebras

Other Feinstein Pages

• For my country by country comparison of peak phone rates to over 55 countries, see my web page Joel Feinstein's Favourite UK Telephone Companies, II
• Elaine Feinstein's Home Page: Information about Elaine Feinstein, poet and novelist, and her work.
• LOOKING UP, THE MONTHLY INTERNATIONAL AUTISM NEWSLETTER Edited by Adam Feinstein, LOOKING UP is a monthly 40-page international newsletter devoted to autism. Aimed at parents and professionals alike, it is a lively and informative mix of the very latest findings in autism research from around the world and articles which explore the impact of autism in all areas.

If you want to send me email, my address is Joel@feinst.demon.co.uk

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