SKIP PAST THE ANNOUNCEMENTS AND WELCOME LECTURE if you so desire. ._. (or to near the bottom)
Crudely speaking, one can say that linear algebra is the study of finite-dimensional, flat things (vector spaces and the linear operators that act on them), while functional analysis studies infinite-dimensional, flat things and differential geometry studies finite-dimensional, curved things. They are the natural generalizations, of which linear algebra is the foundation.
However, it is not quite correct to say that linear algebra is just finite-dimensional; it deals with those things that are true of all vector spaces, finite or infinite in dimension. But those purely algebraic concepts don't get one very far in studying infinite-dimen-sional vector spaces. Topological concepts (convergence) need to be introduced. That is what makes functional analysis. One of the main goals of this course is to move beyond the finite-dimensional focus of elementary linear algebra courses into the setting of infinite dimensions.
As far as linear algebra, in the strict sense, is concerned, this course will be ``in principle, rigorous''. That is, all concepts will be precisely defined, and the proofs of theorems will be either
On the other hand, in discussing infinite-dimensional spaces it will often be necessary to allude to convergence concepts. Necessarily there will be some vagueness and handwaving at these points. They will be included in the lectures for orientation and motivation, with the understanding that a proper treatment is left for later courses in functional analysis.
It is assumed that everyone here has had a first course in linear algebra. Our emphasis is on things that are usually not covered in the undergraduate courses. Nevertheless, the first few weeks will spent in reviewing elementary linear algebra -- partly because people tend to forget things, but also to take the opportunity to generalize and deepen your understanding of the basics. For example, I want to underline that most things learned for finite-dimensional real vector spaces are also true of infinite-dimensional or complex spaces.
We'll start with some homework (see below) to review the elementary calculations with matrices; no formal lectures on that should be necessary, but I (and your classmates, I hope) will be happy to answer questions on eCampus. At this point my old notes say, "More advanced computations (determinants, row reduction) will be reviewed later, when the need arises," but I can't see anywhere in the notes where that promise was fulfulled. Bowen & Wang seem to have never heard of row reduction (Gaussian elimination), which is essential for doing hand calculations with matrices efficiently. Let me know if you want me to go into that in more detail. Meanwhile you can look at Chapter 2 of my undergraduate textbook (see link above).
If you have not yet looked at our eCampus page, please do so now. Click on "Discussions" in the menu on the left, then on the Forum title, "Let's get acquainted". To add your own short bio to the forum, click on "Create Thread". (You have the power to create new threads, but not to create new forums.) To reply to somebody else's post, click on "reply" within that post rather than making a new thread. I'll make a new forum every week or two to hold substantive discussions about the math.
In both lecture notes and homework you'll frequently see references to "Milne". That means the book (good, but now out of print) by R. D. Milne, Applied Functional Analysis: An Introductory Treatment.
Lecture notes on basic definitions about vector spaces.
Homework 1 (due Wednesday, Sept. 5) Instructions for how to submit the papers will be forthcoming soon.
Lecture notes on bases. (These will take us into next week.)
Lecture notes on direct sums.
Homework 2 (due Wednesday, Sept. 12)
Lecture notes on inner products.
Lecture notes on orthogonality.
Homework 3 (due Wednesday, Sept. 19) Henceforth, references otherwise unidentified will always be to Bowen and Wang's book.
Lecture notes on linear operators.
Lecture notes on kernels and ranges.
Lecture notes on isomorphisms and projections.
Homework 4 (due Wednesday, Sept. 26) The last two problems set up an example of Fredholm theory involving differential equations, to be exploited next week.
Lecture notes on adjoints and Fredholm operators.
Homework 5 (due Wednesday, Oct. 3)
Lecture notes on domain technicalities. (Necessary for full disclosure; don't worry about it this semester.)
Lecture notes on hermitian and unitary operators.
Lecture notes on preliminaries to spectral theory. (These will take us into next week.)
Prepare for a test next weekend (Oct 12-14).
Homework 6-7 (due Wednesday, Oct. 17)
Lecture notes on the finite-dimensional spectral theorem.
Lecture notes on an alternative proof.
If you have time, start reading next week's notes on factor spaces.
Homework 8 (due Wednesday, Oct. 24)
Lecture notes on factor spaces
Lecture notes on elementary de Rham cohomology as another example of factor spaces (A few of the diagrams in these notes are still not finished. Here they are in pen.)
Homework 9 (due Wednesday, Oct. 31)
Lecture notes on Jordan canonical form
Lecture notes on proof of the Jordan theorem (Look here for the diagrams that are missing handwritten annotations.)
Homework 10 (due Wednesday, Nov. 7) (Election Day special -- not in the sense that you can wait till Wednesday to vote, but that the assignment is so short you will have plenty of time to vote on Tuesday)
Lecture notes on functions of an operator
Lecture notes on other applications of Jordan theory
Homework 11 (due Wednesday, Nov. 14)
Lecture notes on antisymmetric operators and related matter (A missing diagram is here.)
Homework 12 (due Wednesday, Nov. 28 [sic])
Lecture notes on linear functionals and dual spaces
Homework 13 (due Wednesday, Dec. 5)
Lecture notes on bilinear forms
Lecture notes on general tensors (The missing diagrams are here.)
Go to home pages: Fulling ._._. Calclab ._._. Math Dept ._._. University
e-mail: fulling@math.tamu.edu