Lecture notes for Math 55a: Honors Advanced Calculus and Linear Algebra (Fall 1999)

If you find a mistake, omission, etc., please let me know by e-mail.

The orange balls mark our current location in the course, and the current problem set.

h1.ps: Ceci n'est pas un Math 55a syllabus.

top1.ps: Metric Topology I, basic definitions and examples (the metric spaces Rn and other product spaces; isometries; boundedness and function spaces)

top2.ps: Metric Topology II, open and closed sets and related notions

top3.ps: Metric Topology III, introduction to functions and continuity

top4.ps: Metric Topology IV, sequences and convergence etc.

top5.ps: Metric Topology V, compactness and sequential compactness

a bit of Hausdorff stuff

top6.ps: Metric Topology VI, Cauchy sequences and related notions (completeness, completions, and a third formulation of compactness)

at least in the beginning of the linear algebra unit, we'll be following the Axler textbook closely enough that supplementary lecture notes should not be needed. Some important extensions/modifications to the treatment in Axler:

Here are some practice problems for the final exam, covering both topology and linear algebra (sometimes both in the same problem). These are intentionally harder than I expect the final to be.

The following remarks concerning ``little o notation'' are relevant to the concepts of differentiability etc.; cf. the first ``Remark'' in Rudin, p.213. If f,g are functions on the same metric space, the notation ``f=o(g) as x approaches x0'' means: for every positive epsilon there is a neighborhood of x0 on which |f(x)| <= epsilon g(x). Note that for this to make sense, g had better be a nonnegative real function, and f must take values in a normed vector space -- though equivalent norms yield the same meaning for ``f=o(g)''. Thus: F has derivative F'(x) at x if and only if F(x+h)=F(x)+F'(x)h+o(|h|) as h approaches 0. (Here F is a vector-valued function defined on some neighborhood of x.) The advantage of this is that we don't have to fiddle with the special case h=0.

Check that: if f=o(h) and g=o(h) then f+g=o(h); if f=o(h) and g is a bounded scalar-valued function then fg=o(h); if f=o(g) and g=o(h) then f=o(h); if x is a function of y continuous at y0, and f=o(g) as x approaches x(y0), then f=o(g) as y approaches y0. The chain rule for vector-valued functions (Rudin., p.214, Theorem 9.15) then becomes clear.

[Incidentally, there's also a ``big O'' notation: ``f=O(g)'' means ``there exists a constant C such that |f(x)| <= C g(x) for all x''. Note that this is conceptually simpler since there is no choice of epsilons.]

p1.ps: First problem set: Metric topology

p2.ps: Second problem set: Metrics, topologies, continuity, and sequences

p3.ps: Third problem set: Sequences cont'd; compactness start'd

p4.ps: Fourth problem set: Completeness, and compactness: the grand finale

p5.ps: Fifth problem set: Vector space basics

p6.ps: Sixth problem set: Bases, dimension, etc.

p7.ps: Seventh problem set: Linear maps; duality and adjoints; a bit about field extensions.

p8.ps: Eighth problem set: Duality, cont'd; eigenstuff; a bit about bilinear forms and general norms.

p9.ps: Ninth problem set: Inner-product spaces, and more about duality.

p10.ps: Tenth problem set: more about inner products; normal and self-adjoint operators; a bit about permutations and determinants.