Tuesday 28th September, 16:00-17:00. Administrative
matters. Roughly, the "metric spaces" we
are going to study in this module are sets on which a distance
is defined on pairs of points. This distance function will satisfy
a minimal set of axioms.
One motivation for doing this
is to extend definitions and results from the analysis
of functions
of a single real variable (the topic of the Convergence and Continuity
module) to a more general setting. We considered three concepts
from C&C - convergence of sequences, continuity of functions and
completeness of the reals - and saw how these could be expressed
in metric space terms (informally, since we haven't actually defined
"metric space" yet).
Thursday 30th September, 09:00-10:00.
We saw on Tuesday that some of the concepts from C&C can
be generalised. Now, a couple of specimen theorems we would
like to generalise: Every sequence in [a,b] has a convergent
subsequence. Every cts real function
on a closed interval [a,b] is bdd. Axioms for a metric, and
some examples: "usual" metric on R and path metric on graphs.
Proofs that these are metrics. A couple of non-examples.
Friday 1st October, 13:00-14:00.
[Exercise Sheet 1 released.]
Definition of the Manhattan metric (ell^1) on R^n and a proof that
it is a metric. [Break to consider a couple of questions from
last time.]
Definition of the Euclidean metric (ell^2) on R^n and a proof that
it is indeed a metric.
Tuesday 5th October, 15:00-16:00.
The ell^infty metric on R^n. (Verification that it is a metric will
be an exercise on Sheet 2).
Function spaces; the "sup metric"
(or "uniform metric") on bdd real functions from a set S.
Proof that the sup metric is a metric.
Thursday 7th October, 09:00-10:00.
Definition of metric space, subspace (which is itself a metric
space).
Informal d Example: C[a,b] is the
set of continuous functions on the closed interval [a,b] with the
sup-norm. Then C[a,b] is a subspace of B[a,b], since every continuous
function on a bounded closed interval is bounded.
Discussion of another metric on function spaces (L^1 distance).
Definition of norm ||.|| on a vector space and of a normed
space. Every normed space can be viewed as a metric space with
metric defined by rho(x,y) = ||x - y||.
Friday 8th October, 13:00-14:00.
[Exercise Sheet 2 released.]
Example: ell^infty norm (and hence metric) on R^n.
Discussion: we have met three
metrics on R^n: ell^1 (Manhattan), ell^2 (Euclidean) and ell^infty
(sup); these can all be obtained from the corresponding normed space.
Three ways to construct product spaces. Proof that first of them
is correct (does define a metric). Construction of ell^1, ell^2 and
ell^infty metrics as product spaces.
One final example of a metric: the discrete metric (important as
a counterexample!).
Monday 11th October, 11:00-12:00.
New topic: convergence.
Definition of convergence of a sequence in a metric space
to a point in that space. Example: x_n = (n^{-1}cos(an),
n^{-1}sin(an)) converges to (0, 0) in (R^2, d_1), (R^2, d_2)
and (R^2, d_infty), but not in the discrete metric.
Definition of open and closed balls in a metric
space. Example: unit balls in (R^2, d_1), (R^2, d_2),
(R^2, d_infty).
Thursday 14th October, 09:00-10:00.
Alternative definition of convergence in terms
of eventual membership of elements of the sequence in balls of
arbitrarily small radius. Definition of equivalent metrics
(in terms of convergence of
sequences). Example: d_1 (Manhattan) and d_2 (Euclidean)
are equivalent metrics on R^n. (Convergence
in d_1 implies convergence in d_2. The reverse direction is a question on
Sheet 3). Review of Q4 on Exercise Sheet 1.
Friday 15th October, 13:00-14:00.
[Exercise Sheet 3 released.]
Alternative
characterisation of equivalence of metrics rho and sigma
(every rho-ball contains some sigma-ball and vice versa).
Proof of equivalence with earlier definition. Example:
d_2 and the discrete metric on R^2 are not equivalent.
Monday 18th October, 11:00-12:00.
Example: In every open ball in the d_infty metric there is a Euclidean
open ball. (Opposite direction is in Exercise Sheet 3. Conclusion
is that d_infty and d_2 are equivalent metrics.)
Uniform convergence as convergence f_n -> f in B(S). Example:
convergence of polynomial approximations to exp(x)
on [0,1].
Characterisation of uniform
convergence in terms of: "For all epsilon > 0, there exists N_epsilon
such that ...", as per D&IA.
Proof of equivalence of these two views.
Thursday 21st October, 09:00-10:00.
Pointwise convergence
of functions. Comparison with uniform
convergence in B(S). Various
illustrative
examples from the "Flash&Math" site.
Example: the "narrowing spike" function on [0,1].
Friday 22nd October, 13:00-14:00.
[Exercise sheet 4 released; notes on sheet 1 distributed.]
Informal comparison with convergence in L^1 metric on C[0,1].
New topic:
open and closed sets. Defintion of open set in a metric
space (X, rho). Lemma: the open ball is open.
Examples and non-examples, from the real line.
Monday 25th October, 11:00-12:00.
Thm. The union of open sets is open, the intersection of a finite
collection of open sets is open.
The properties of open sets that we proved at the end of last
time can be turned round and made into a definition. A topological
space is a collection of open sets having those properties.
The theory of topological spaces is more general than that of
metric spaces: not every topological space in "metrizable".
Thursday 28th October, 09:00-10:00.
[Notes on sheet 2 distributed.]
Lemma: A set is open iff it is the union of a collection of open balls.
Definition of the interior of a set. Examples.
Definition of a closed set.
Examples. A closed ball is a closed set.
Review of Q2 on Exercise Sheet 3.
Friday 29th October, 13:00-14:00.
[Exercise sheet 5 released.]
Definition of limit point. Examples
and non-examples. Alternative characterisation of closed set: a set is
closed if it contains all its limit points. Proof of
equivalence with definition of closed set.
The whole space X and emptyset are closed; intersections of
arbitatry collections of closed sets are closed; and
unions of finitely many closed sets are closed.
An example of an infinite union of closed sets that
is not closed: intervals [-1+1/n, 1-1/n] in R.
Definition of closure of a set.
Monday 1st November, 11:00-12:00.
Review of continuity of real functions from C&C.
Definition from C&C generalises easily to maps (X, rho) ->
(Y, sigma), where (X, rho), (Y, sigma) are
arbitrary metric spaces.
Epsilon-delta definition of continuous function recast in
terms of open balls. Ex. The function f(x,y) = (x^2 + y^2, xy) is
continuous at (0,0), where we assume the Euclidean metric on R^2.
[Time allowed to complete questionnaire.]
Thursday 4th November, 09:00-10:00.
[Notes on Sheet 3 distributed.]
Equivalent characterisation of continuous
function in terms of convergent sequences. Proof of equivalence.
Review of Q3(a) from Ex Sheet 4.
Example: function
of two real variables, f(x,y) = xy/(x^2 + y^2) (to be continued).
Friday 5th November, 13:00-14:00. [Exercise sheet 6 released.]
Example (continued):
f(x,y) = xy/(x^2 + y^2)is continuous in x and y separately, but not jointly.
Example: The map I: C[0,1] -> R defined by I(f) is the integral
from x equals 0 to 1 of f(x) is continuous.
Theorem: composition of continuous functions is continuous. Proof
using characterisation in terms of convergent sequences.
Discussion of Q3(b) from sheet 4.
Week 7, beginning 8th November.
No lectures this week.
Monday 15th November, 11:00-12:00.
Definition of inverse image f^{-1}(A) of a set A. Several examples
based on f(x) = sin x. Note that, in our examples, the inverse image
of an open set is open, and that of a closed set is closed.
Theorem: the following are equivalent: (i) a function is continuous,
(ii) the inverse image of every open set is open, and (iii) the
inverse image of any closed set is closed. Proof.
Thursday 18th November, 09:00-11:00.
[Notes on Sheet 4 distributed.]
Example
of a discontinuous function, demonstration that is discontinuous
against the the three characterisations of continuity
considered in the course.
Remark: Equivalent metrics lead to the same collection of
continuous functions.
New topic: complete metric
spaces. Definition of Cauchy sequence. Lemma. A convergent
sequence is Cauchy. Converse? No, e.g., rationals with usual metric.
Definition of a complete metric space.
Non-example: Q.
Friday 19th November, 13:00-14:00.
[Exercise sheet 7 released.]
Examples: R and Z.
Theorem: Every metric space has a completion.
Sketch proof of theorem.
Discussion of Q3(b) from sheet 5.
Monday 22nd November, 11:00-12:00.
Example: completion of (0,infty).
Theorem: A closed subset of a complete metric space is complete.
Theorem: B(S) is complete. Proof.
Step 1: a Cauchy sequence f_n in B(S) converges pointwise to some
function f: S -> R. Step 2. Show that convergence is actually
uniform. Step 3. Show that f is bounded (deferred).
Thursday 25th November, 09:00-10:00.
[Notes on sheet 5 distributed.]
Step 3. Show that f is bounded.
C[a,b] is complete. Proof.
Defintion of contraction on a metric space (X, rho).
Lemma: A contraction on a metric
space is continuous.
Friday 27th November, 13:00-14:00.
[Exercise sheet 8 released.]
Contraction Mapping Theorem: if f
is a contraction on a complete metric space then
f(x) = x has a unique solution,
and a, f(a), f(f(a)), f(f(f(a))), ... converges to x
for all a in X.
Corollary: form of the error bound.
Monday 29th December, 16:00-17:00.
Application 1. f: R -> R given by f(x) = x/2 + 1/x is a contraction
on [1, infty). The x_0 = 1, x_1 = f(x_0) = 1.5, x_2 = f(x_1) =
1.4167-, x_3 = f(x_2) = 1.4142+ converges to sqrt(2), the unique
solution to f(x) = x in the range [1, infty).
Application 2. Find solution to phi'(x) =
alpha phi(x), phi(0) = 1 in C[0, b] (b to be determined).
Recast as integral equation using fundamental theorem of calculus.
Write down map T : C[0,b] -> C[0,b] such that integral
equation becomes T(phi) = phi. Show T is a contraction
provided b < 1/|alpha|. Start with phi_0 = 0; then
phi_1 = T(phi_0) = 1, phi_2 = T(phi_1) = 1 + alpha x,
phi_3 = T(phi_2) = 1 + alpha x + (alpha x)^2/2, etc.
These are truncations of the Taylor series expansion of exp(x).
So phi_n -> exp in C[0,b].
(Calculation was omitted.)
Example of a differential equation without a unique solution
(y' = (3/2)y^(1/3)).
Thurday 2nd December, 16:00-17:00.
No lecture.
Friday 3rd December, 09:00-10:00.
New topic: compactness. Motivation, with reference to
results from C&C. Subsequence of a sequence. Definition
of (sequential) compactness. Example (2-point space) and non-examples
(R and (0,1)).
Definition of a bounded set. Theorem: a compact set is
closed and bounded. Proof that not closed => not compact.
Monday 6th December, 11:00-12:00.
Proof that not bounded => not compact.
A closed subset of a compact set is compact.
Proof.
Examples: (0,1), [0,infty) and {0,1,2} as subsets of R
(with the usual metric). Lemma: The Cartesian product of compact sets
is compact in the product metric. Discussion of proof strategy.
Thursday 9th December, 09:00-10:00.
[Notes on sheet 6 distributed.]
Proof of lemma from last time.
Lemma: A closed interval of the real line is compact.
Proof by bisection.
Friday 10th December, 13:00-14:00.
[Exercise sheet 9 released.]
Theorem:
Any closed and bounded subset K of R^n is compact in the
Euclidean metric (Heine-Borel). Proof. K is contained
in a closed cube Q^n for a sufficiently large interval Q.
Q^n is a product of compact sets and thus compact.
K is a closed subset of a compact set and hence compact.
Theorem: The image of a compact set under a continuous map
is compact. Proof. Counterexample:
the image of a compact set under a discontinuous
function may not be compact. Review of Q1 from sheet 7.
Monday 13th December, 11:00-12:00.
Technical lemma: If a Cauchy sequence
has a convergent subsequence then the sequence itself is convergent,
and to the same limit.
Definition of a compact metric space.
Theorem. A compact metric space is complete.
A real function on a compact metric space is bounded
and attains it infimum and supremum. An alternative
definition of "compact".
Thursday 16th December, 09:00-10:00.
[Notes on sheet 7 distributed.]
Examples of non-compact spaces against definition in terms of
open covers.
Definition of uniformly continuous function. Comparison
with definition of continuous function.
Theorem: A continuous
function on a compact metric space is uniformly
continuous. Proof.
Friday 17th December, 13:00-14:00.
Examples of functions that are continuous but not
uniformly continuous. We know that
a compact metric space is complete and bounded.
An example of a metric space that is complete
and bounded but not compact. Brief discussion
of total boundedness.
