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 **R**^{n}
and other product spaces; isometries; boundedness and function spaces

IfSis an infinite set andXis an unbounded metric space then we can't use our definition ofX^{S}as a metric space because sup_{S}d(_{X}f(s),g(s)) might be infinite. But thebounded functionsfromStoXdoconstitute a metric space under the same definition ofd. A function is said to be ``bounded'' if its image is a bounded set. You should check that that_{XS}d(_{XS}f,g) is in fact finite for boundedfandg.

The ``Proposition'' on page 3 of the first topology handout can be extended as follows:Metric Topology II (PS, PDF)iv) ForIn other words, for everyeverypointpofXthere exists a real numberMsuch thatd(p,q)<Mfor allqofE.pinXthere exists an open ball aboutpthat containsE. Do you see why this is equivalent to (i), (ii), and (iii)?

Open and closed sets, and related notions

p.4: closure of

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)

at least in the beginning of the

- [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**/*p***Z**(*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**_{p}(*p*prime), introduced at the end of our topology unit; and more exotic finite fields such as the 9-element field (**Z**/*3***Z**)(*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*_{1}*v*_{1}+ ... +*a*with each_{n}v_{n}*v*in_{i}*V*and each*a*in_{i}*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}=*V*_{0},*V*_{1},*V*_{2}, ...,*V*, with each_{n}=V*V*of dimension_{i}*i*and containing*V*_{i-1}. A basis*v*_{1},*v*_{2}, ...,*v*determines a flag:_{n}*V*is the span of the first_{i}*i*basis vectors. Another basis*w*_{1},*w*_{2}, ...,*w*determines the same flag if and only if each_{n}*w*is a linear combination of_{i}*v*_{1},*v*_{2}, ...,*v*(necessarily with nonzero_{i}*v*coefficient). The_{i}*standard flag*in*F*is the flag obtained in this way from the standard basis of unit vectors_{n}*e*_{1},*e*_{2}, ...,*e*. The punchline is that, just as a diagonal matrix is one that respects the standard basis (equivalently, the associated decomposition of_{n}*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 automorphismsigma: such that sigma is not the identity but sigma*F*-->*F*^{2}is (that is, sigma is an involution). Given a basis {*v*_{i}} for*F*, a sesquilinear form <.,.> on*F*is determined by the field elements*a*_{i,j}=<*v*_{i},*v*_{j}>, and is conjugate-symmetric if and only if*a*_{j,i}=sigma(*a*_{i,j}) 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*U*_{1},*U*_{2}for which the pairing is positive definite on*U*_{1}and negative definite on*U*_{2}. Then any subspace*W*of*V*on which the pairing is positive definite has dimension no greater than dim(*U*_{1}). Proof: On the intersection of*W*with*U*_{2}, 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*) for some nonzero scalars_{i}x_{i}y_{i}*a*. But in general it can be quite hard to decide whether two different collections of_{i}*a*yield isomorphic pairings. Even over_{i}**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*=0;^{k}

(2) For any vector*v*, there exists a nonnegative integer*k*such that*T*=0;^{k}v

(3)*T*=0, where^{n}*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*).

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.

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.

First & second problem set: Metric topology (PS, PDF)

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 ||

also in 7, a sentence to say explicitly that

and in the introductory paragraph for 8, the pairing is on

and (16.xii.05) again in 8, the ``associated pairing'' is now specified.