Suppose we consider functions defined on some nice set ,
for example . Then it is useful to have an idea how
large the functions
are

**Definition 1:**

A *norm* on a vector space **V** is a real valued function on **V**, satisfying

- a)
- for all , and .
- b)
- for all .
- c)
- for all and .

A space is called a *metric space* provided there is
defined a real-valued distance between every , such that

- a)
- for all
- b)
- for every ,
- c)
- whenever
- d)
- for all

(limits in a normed space)

**Definition 4:**

A sequence is called a Cauchy sequence in if an only if for every there exists an integer such that

``In second year calculus we learn that every Cauchy sequence
converges, but that is for
sequences in , not any old space!'' *M.A. McKiernan
(Applied Math.,
Waterloo)*

**Definition 5:**

is a complete metric space if every Cauchy sequence
converges to a limit in

**Definition 6:**

A normed linear space that is complete when considered as a metric
space is called a *Banach
space*

**Definition 7:**

Let **X** denote a linear space over (or over ). An
*inner product* is a
mapping satisfying

- where ``--------'' denotes complex conjugate
- and iff
**x = 0**.

Show that the function defines a norm, so we may write

A Hilbert space is an inner product space which, with the norm , is a
Banach space

**Definition 9:**

A Sobolev norm is a norm of the form

or similar involving higher derivatives (the idea is that the
derivatives are
involved)

**Remark:** if **p = 2** this is an inner product
norm

**Definition 10:**

A *Sobolev space* is a Banach space where the metric is
given by
where
is a Sobolev norm

Pages 22-26 Sobolev: key points

- Cauchy problem:
**Lu = 0**in (infinite) - Dirichlet problem:
**Lu = 0**in - Neumann problem:
**Lu = 0**in

which holds when , and the boundary conditions are given on certain manifolds by

where is a given function defined on the manifold .
We now consider Sobolev
norms on functions defined on the , and a Sobolev norm on
functions (in a class **u**)
defined on .

and

For **u** we take the class of functions with continuous derivatives up
to order **p** in ,
and for we take the class of functions with continuous
derivatives (with respect to the
parameters of the manifolds ) up to order **k**. For the Sobolev
norms we take, eg.,

and

means all possible j order derivatives

(this is the -norm case of the norms given before). Then our
problem *depends
continuously on the data* provided that for all there
exists a such that if

then

where solves **Lu = 0** in , on

and **u** solves **Lu = 0** in , on
. A problem is
called *well-posed* if

- (a)
- a solution exists
- (b)
- the solution is unique
- (c)
- the solution depends continuously on the data.

Consider the problem (ODE)

and the neighbouring problems

and discuss whether it is likely that this could be well-enough conditioned. (It is, like all smooth IVP for ODE, well-posed.)

Wed Feb 4 14:46:04 EST 1998