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.

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:

• [cf. Axler, p.3] Unless noted otherwise, F may be an arbitrary field, not only R or C. The most important fields other than those of real and complex numbers are the field Q of rational numbers, and the finite fields Z/pZ (p prime). Other examples are the field Q(i) of complex numbers with rational real and imaginary parts; more generally, Q(d^1/2) for any nonsquare rational number d; the ``p-adic numbers'' Q<sub>p</sub> (p prime), introduced at the end of our topology unit; and more exotic finite fields such as the 9-element field (Z/3Z)(i). Here's a review of the axioms for fields, vector spaces, and related mathematical structures.
• [cf. Axler, p.22] We define the span of an arbitrary subset S of (or tuple in) a vector space V as follows: it is the set of all (finite) linear combinations a<sub>1</sub> v<sub>1</sub> + ... + a<sub>n</sub> v<sub>n</sub> with each v<sub>i</sub> in V and each a<sub>i</sub> in F. This is still the smallest vector subspace of V containing S. In particular, if S is empty, its span is by definition {0}. We do not require that S be finite.
• Axler does not seem to explicitly use the important notion of a quotient vector space. If U is a subspace of a vector space V, we get an equivalence relation on V by defining two vectors v,v' to be equivalent (``congruent mod U'') if v-v' is in U. The set of equivalence classes then itself becomes a vector space (this must be proved!), called the quotient space V/U. [Note that this is also a notation for V being a vector space over a field U -- we shall strive to make it clear which meaning we intend when both meanings might make sense.]
• Axler proves the Fundamental Theorem of Algebra using complex analysis, which cannot be assumed in Math 55. Here's a proof using the topological tools we developed in the first month of class. (Axler gives the complex-analytic proof on page 67.)
• Axler unaccountably soft-pedals the important notion of duality; we devote much of the seventh problem set to this, and return to it often later.
• We shall need some ``eigenstuff'' also in an infinite-dimensional setting, so will not assume that any vector space is (nonzero) finite dimensional unless we really must.
• Triangular matrices are intimately related with ``flags''. A flag in a finite dimensional vector space V is a sequence of subspaces {0}=V₀, V₁, V₂, ..., V_n=V, with each V_i of dimension i and containing V_i-1. A basis v₁, v₂, ..., v_n determines a flag: V_i is the span of the first i basis vectors. Another basis w₁, w₂, ..., w_n determines the same flag if and only if each w_i is a linear combination of v₁, v₂, ..., v_i (necessarily with nonzero v_i coefficient). The standard flag in F_n is the flag obtained in this way from the standard basis of unit vectors e₁, e₂, ..., e_n. The punchline is that, just as a diagonal matrix is one that respects the standard basis (equivalently, the associated decomposition of V as a direct some of 1-dimensional subspaces), an upper-triangular matrix is one that respects the standard flag. Note that the i-th diagonal entry of a triangular matrix gives the action on the one-dimensional quotient space V_i/V_i-1 (each i=1,...,n).
• As explained in class, the generalization to arbitrary fields F of Axler's treatment (pages 91-93) of invariant subspaces on real vector spaces looks like this:
  Suppose u is a nonzero vector in an F-vector space V, and Q is an irreducible polynomial of degree d>0 in F[X] such that (Q(T))u=0. Then the d vectors u,Tu,T²u, ..., T^d-1u span a T-invariant subspace U of V. Our key claim was that these vectors are in fact linearly independent, and thus that U is d-dimensional. Indeed, a purported linear relation would take the form (R(T))u=0 for some polynomial R, not identically zero, of degree strictly less than d. But then Q and R are relatively prime, so there exist polynomials A,B such that AQ+BR=1. Thus A(T)Q(T)+B(T)R(T) is the identity operator on V. Evaluating at u yields 0=u, contradiction.

  To put this AQ+BR trick in context: let I_u be the set of polynomials P such that P(T)u=0. Then I_u is an ideal in F[X]: if P,P' are in I_u then so is P+P', as well as AP for any polynomial A. Now for any polynomial Q, the set of all polynomials of the form AQ is an ideal, the so-called ``principal ideal'' generated by Q. Conversely, every ideal in F[X] is principal. This is proved as it is for ideals in Z, using the division algorithm (Axler, 4.5 on page 66 ff.). This means than if an ideal I contains an irreducible polynomial Q -- and thus contains the principal ideal generated by Q -- then either I is that principal ideal, or it is all of F[X]. Going back to our case I=I_u, we find that in the first case U has dimension d, and in the second case I_u contains 1 so u=0. [Alternatively, given Q and R, the set of all polynomials of the form AQ+BR is again an ideal, etc.]

• A bilinear pairing on a vector space V is said to be skew-symmetric (a.k.a. ``anti-symmetric'' or ``alternating'') if <x,x>=0 for all x in V. By expanding <x+y, x+y> we deduce that <x,y> = - <y,x> for all x, y in V, whence the terminology. Conversely the identity <x,y> = - <y,x> implies <x,x>=0 (let y=x), except when the ground field F has characteristic 2, in which case <x,y> = - <y,x> amounts to the condition that the pairing be symmetric.
• A norm on a real or complex vector space V is any function v |--> |v| from V to the real numbers such that:
  • The identity |cv| = |c| |v| holds for all scalars c and all vectors v;
  • d(x,y) := |x-y| defines a distance function on V.
  The second condition means that: |v| is nonnegative for all v, and zero iff v=0; and |v+v'| <= |v| + |v'| for all vectors v, v'. (These correspond to the nonnegativity and triangle inequality respectively; what happened to symmetry?) For instance, we have seen in effect that |x| := max(|x₁|, ..., |x_n|) defines a norm on Rⁿ or Cⁿ; on pages 102-105, Axler in effect verifies that |x|² = (x,x) gives a norm on an inner-product space.
• Here's an outline of the standard proof of the Spectral Theorem for self-adjoint oprators on a finite-dimensional real inner product space.
• Here are basic definitions and constructions involving groups that give a more general context for some of our results so far and will be especially useful in our coming discussion of determinants.
• Here are definitions and basic results concerning cofactors and minors of a matrix.
• And here are some symple results and pfacts about symplectic spaces and the Pfaffian.

Here are some

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 x<sub>0</sub>'' means: for every positive epsilon there is a neighborhood of x<sub>0</sub> 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.]

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.