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.

Our first topic is the topology of metric spaces, a fundamental tool of modern mathematics that we shall use mainly as a key ingredient in our rigorous development of differential and integral calculus. To supplement the treatment in Rudin's textbook, I wrote up 20-odd pages of notes in six sections; copies will be distributed in class, and you also may view them and print out copies in advance from the PostScript or PDF files linked below.

Metric Topology I (PS, PDF)
Basic definitions and examples: the metric spaces Rⁿ and other product spaces; isometries; boundedness and function spaces

If S is an infinite set and X is an unbounded metric space then we can't use our definition of X^S as a metric space because sup_S d_X(f(s),g(s)) might be infinite. But the bounded functions from S to X do constitute a metric space under the same definition of d_X^S. A function is said to be ``bounded'' if its image is a bounded set. You should check that that d_X^S(f,g) is in fact finite for bounded f and g.

The ``Proposition'' on page 3 of the first topology handout can be extended as follows:

iv) For every point p of X there exists a real number M such that d(p,q)<M for all q of E.

In other words, for every p in X there exists an open ball about p that contains E. Do you see why this is equivalent to (i), (ii), and (iii)?

Metric Topology III (PS, PDF)
Introduction to functions and continuity
corrected 25.ix.05 (V for Z five times in p.3, paragraph 2)

Metric Topology IV (PS, PDF)
Sequences and convergence, etc.

Metric Topology V (PS, PDF)
Compactness and sequential compactness
corrected 1.x.05 to fix various minor typos

Metric Topology VI (PS, PDF)
Cauchy sequences and related notions (completeness, completions, and a third formulation of compactness)

• [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.
• About the infamous ``Lemma 3.?'': some notes and warnings about the behavior of Hom(V,W) under (finite or infinite) direct sums.
• Unlike Axler, we spend some time on ``quotient vector spaces''.
• Axler also unaccountably soft-pedals the important notion of duality.
• Here's a brief preview of abstract nonsense (a.k.a. diagram-chasing), and a diagram-chasing interpretation of quotients and duality.
• 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, in PS and PDF. (Axler gives the complex-analytic proof on page 67.)
• 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.
• If T is a linear operator on a vector space V, and U is an invariant subspace, then the quotient space V/U inherits an action of T. Moreover, the annihilator of U in V^* is an invariant subspace for the action of the adjoint operator T^* on V^*. (Make sure you understand why both these claims hold.)
• 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 sum 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).

• One of many applications is the trace of an operator on a finite dimensional F-vector space V. This is a linear map from Hom(V,V) to F. We can define it simply as the composition of two maps: our identification of Hom(V,V) with the tensor product of V^* and V, and the natural map from this tensor product to F coming from the bilinear map taking (v^*,v) to v^*(v).
• Here are some basic facts about general norms on real and complex vector spaces.
• Just as we can study bilinear symmetric forms on a vector space over any field, not just R, we can study sesquilinear conjugate-symmetric forms on a vector space over any field with a conjugation, not just C. Here a ``conjugation'' on a field F is a field automorphism sigma:F-->F such that sigma is not the identity but sigma<sup>2</sup> is (that is, sigma is an involution). Given a basis {v<sub>i</sub>} for F, a sesquilinear form <.,.> on F is determined by the field elements a<sub>i,j</sub>=<v<sub>i</sub>,v<sub>j</sub>>, and is conjugate-symmetric if and only if a<sub>j,i</sub>=sigma(a<sub>i,j</sub>) for all i,j. Note that the ``diagonal entries'' a_i,i --- and more generally <v,v> for any v in V --- must be elements of the subfield of F fixed by sigma.
• ``Sylvester's Law of Inertia'' states that for a nondegenerate pairing on a finite-dimensional vector space V/F, where either F=R and the pairing is bilinear and symmetric, or F=C and the pairing is sesquilinear and conjugate-symmetric, the counts of positive and negative inner products for an orthogonal basis constitute an invariant of the pairing and do not depend on the choice of orthogonal basis. (This invariant is known as the ``signature'' of the pairing.) The key trick in proving this result is as follows. Suppose V is the orthogonal direct sum of subspaces U₁, U₂ for which the pairing is positive definite on U₁ and negative definite on U₂. Then any subspace W of V on which the pairing is positive definite has dimension no greater than dim(U₁). Proof: On the intersection of W with U₂, the pairing is both positive and negative definite; hence that subspace is {0}. The claim follows by a dimension count, and we quickly deduce Sylvester's Law.
• Over any field not of characteristic 2, we know that for any non-degenerate symmetric pairing on a finite-dimensional vector space there is an orthogonal basis, or equivalently a choice of basis such that the pairing is (x,y)=sum_i(a_i x_i y_i) for some nonzero scalars a_i. But in general it can be quite hard to decide whether two different collections of a_i yield isomorphic pairings. Even over Q the answer is already tricky in dimensions 2 and 3, and I don't think it's known in a vector space of arbitrary dimension.
• Re Chapter 7: we'll prove the Spectral Theorem for a self-adjoint or Hermitian operator T on a finite-dimensional real or complex vector space, not a la Axler, but by the usual method of maximizing the ``Rayleigh quotient'' <Tv,v>/<v,v>. Likewise for normal operators on a finite dimensional C-vector space, by maximizing <Tv,Tv>/<v,v>.
• All of Chapter 8 works over an arbitrary algebraically closed field, not only over C (except for the minor point about extracting square roots, which breaks down in characteristic 2); and the first section (``Generalized Eigenvalues'') works over any field.
• We don't stop at Corollary 8.8: let T be any operator on a vector space V over a field F, not assumed algebraically closed. If V is finite-dimensional, then The Following Are Equivalent:
            (1) There exists a nonnegative integer k such that T^k=0;
            (2) For any vector v, there exists a nonnegative integer k such that T^kv=0;
            (3) Tⁿ=0, where n=dim(V).
  Note that (1) and (2) make no mention of the dimension, but are still not equivalent for operators on infinite-dimensional spaces. We readily deduce the further equivalent conditions:
            (4) There exists a basis for V for which T has an upper-triangular matrix with every diagonal entry equal zero;
            (5) Every upper-triangular matrix for T has zeros on the diagonal, and there exists at least one upper-triangular matrix for T.
  Recall that the second part of (5) is automatic if F is algebraically closed.
• The space of generalized 0-eigenvectors (the maximal subspace on which T is nilpotent) is sometimes called the nilspace of T. It is an invariant subspace. When V is finite dimensional, V is the direct sum of the nilspace and another invariant subspace V', consisting of the intersection of the subspaces T^k(V) as k ranges over all positive integers. See Exercise 8.11; This can be used to quickly prove Theorem 8.23 and consequences such as Cayley-Hamilton (Theorem 8.20).
• The dimension of the space of generalized c-eigenvalues (i.e., of the nilspace of T-cI) is usually called the algebraic multiplicity of c (since it's the multiplicity of c as a root of the characteristic polynomial of T), to distinguish it from the ``geometric multiplicity'' which is the dimension of ker(T-cI).

We'll define the determinant of an operator T on a finite dimensional space V as follows: T induces a linear operator T' on the top exterior power of V; this exterior power is one-dimensional, so an operator on it is multiplication by some scalar; det(T) is by definition the scalar corresponding to T'. The ``top exterior power'' is a subspace of the ``exterior algebra'' of V, which is the quotient of the tensor algebra by the ideal generated by {v*v: v in V}. We'll still have to construct the sign homomorphism from the symmetry group of order dim(V) to {1,-1} to make sure that this exterior algebra is as large as we expect it to be, and that in particular that the (dim(V))-th exterior power has dimension 1 rather than zero.

Some more tidbits about exterior algebra:

• If w, w' are elements of the m-th and m'-th exterior powers of V, then ww'=(-1)^mm'w'w; that is, w and w' commute unless m and m' are both odd in which case they anticommute.
• If m+m'=n=dim(V) then the natural pairing from the m-th and m'-th exterior powers to the n-th is nondegenerate, and so identifies the m'-th exterior power canonically with the dual of the m-th tensored with the top (n-th) exterior power.
• In particular, if m=1, and T is any invertible operator on V, then we find that the induced action of T on the (n-1)st exterior power is the same as its action on V^* multiplied by det(T). This yields the formula connecting the inverse and cofactor matrix of an invertible matrix (a formula which you may also know in the guise of ``Cramer's rule'').
• For each m there is a natural non-degenerate pairing between the m-th exterior powers of V and V^*, which identifies these exterior powers with each other's dual.

We'll also show that a symmetric (or Hermitian) matrix is positive definite iff all its eigenvalues are positive iff it has positive principal minors (the ``principal minors'' are the determinants of the square submatrices of all orders containing the (1,1) entry). More generally we'll show that the eigenvalue signs determine the signature, as does the sequence of signs of principal minors. For positive definiteness, we have the two further equivalent conditions: the symmetric (or Hermitian) matrix A=(a<sub>ij</sub>) is positive definite iff there is a basis (v<sub>i</sub>) of F<sup>n</sup> such that a<sub>ij</sub>=<v<sub>i</sub>,v<sub>j</sub>> for all i,j, and iff there is an invertible matrix B such that A=BB<sup>*</sup>. For example, the matrix with entries 1/(i+j-1) (``Hilbert matrix'') is positive-definite, because it is the matrix of inner products (integrals on [0,1]) of the basis 1,x,x²,...,x^n-1 for the polynomials of degree <n. Can you find the determinant of this matrix?

More will be said about exterior algebra when differential forms appear in Math 55b.

Here's a brief introduction to field algebra and Galois theory.

Third problem set: Metrics, sequences, compactness, and a bit of general topology (PS, PDF)

Fourth problem set: Topology grand finale (PS, PDF)

Fifth problem set / Linear Algebra I: vector space basics (PS, PDF)

Sixth problem set / Linear Algebra II: the dimension and some of its uses (PS, PDF)

Seventh problem set / Linear Algebra III: linear maps and duality (PS, PDF)

Eighth problem set / Linear Algebra IV: Eigenstuff, and a projective overture (PS, PDF)
corrected 9.xi.05 (typo in 1/iii)

Ninth problem set / Linear Algebra V: Tensors, etc. (PS, PDF)

Tenth problem set / Linear Algebra VI: Inner products, lattices, and normal operators (PS, PDF)
Here's my solution for problem 4 (and 3) (PS, PDF)

Eleventh problem set / Linear Algebra VII: Fourier foretaste, and symplectic structures (PS, PDF)

Twelfth and last problem set / Linear Algebra VIII: Groups, exterior algebra, and determinants (PS, PDF)
corrected 15.xii.05 (v_i, not ||v_i||, in 6(i); typo in 7 -- thanks to Scott K. for both;
also in 7, a sentence to say explicitly that k is fixed throughout;
and in the introductory paragraph for 8, the pairing is on V^*, not V, and the map is from V^* to V, not V to V^*);
and (16.xii.05) again in 8, the ``associated pairing'' is now specified.