From: george.ivey@gallaudet.edu (G. E. Ivey)
Subject: re: Metric Space- what's it about?
Date: 30 Nov 2000 09:23:39 -0500
Newsgroups: sci.math
Summary: [missing]

Kelly McCauley wrote:
>I'm looking at taking a graduate level class - "Metric spaces with
>applications".  I have not come across this math object before and
>I'm wondering what the topic will be like.  I know the definition
>and such, but  I'm looking for some hint or advice on what to expect.

>I've recently completed a linear algebra course (vector spaces, inner
>products, diagonalizaion, eigenvalues, etc.) and it looks like this
>is a continuation of that sort of topic.

     It might be correct to say "that sort".  The concept of metric
space lies (in abstraction) between very concrete things like
Euclidean Space (R^n) and more abstract topology.  
     A "metric function" is a function that assigns a "distance" to
every pair of points.  The requirements for any metric function are:
1) f(x,x)= 0 and f(x,y)>0 if x is not equal to y. (The "distance"
between any point and itself is 0 while the "distance" between
distinct points is positive.)

2) f(x,y)= f(y,x).  (The "distance" from x to y is the same as the
"distance" from y to x!)

3) For any three points, x,y,z, f(x,y)< or = d(x,z)+ d(z,y).  (This is
called the "triangle inequality".  Draw a picture with x,y, z points
in the plane and think about the straight line distances involved.)

    Of course, if we  are thinking of points in a plane (or 3
dimensions), and measure the straight line distance between points,
that satisfies these requirements and is a metric function.  I put
"distance" in quotes above because there are many other functions
satisfying these.  In R^3 for example, we can define the distance from
(x1,y1,z1) to (x2,y2,z2)to be sqrt((x2-x1)^2+(y2-y1)^2+(z2-z1)^2)
(the "usual" metric) or |x2-x1|+ |y2-y1|+ |z2-z1|  or max(|x2-x1|,
|y2-y1|,|z2-z1|).  One of the fundamental results of "metric spaces"
is that the topology defined by these three (neighborhoods and open
sets so we can talk about "closeness") are exactly the same.  
    On the other hand, we can also "measure the distance" between two
functions, f and g.  One way is d(f,g)= sqrt(integral(f(x)-g(x))^2dx
(The "L2" metric).  Another is d(f,g)= integral |f(x)-g(x)| dx (the
"L1" metric).  Yet another is d(f,g)= max|f(x)-g(x)| (the "uniform"
metric).  You will notice that these are analogous to the three given
for R^n.  An important difference is that, since the function spaces
are infinite dimensional, these three metrics give very different
"topologies" and, so, different limit and convergence properties.
    These particular metrics are especially important in functional
analysis.  A basic difference between Linear Algebra and metric spaces
is that metric spaces do not have the algebraic structure:  we
"measure the distance" between two "points" but don't add or subtract.
    You might remember "inner product" from linear algebra.  Although
a vector space does not necessarily have an inner product, if we do
have one we can then define a "norm" by ||v||= sqrt(<v,v>.  Once we
have a norm, we can define a metric by d(u,v)= ||u-v||.  And, like I
said, once we have a metric we can define "neighborhoods" and,so, a
topology.
    These are different levels of abstraction because there exist
topologies that do not have metrics associated with them, there exist
metric spaces which do not come from norms and there exist norms which
are not defined by inner products.

    Basically, we need a "topology" to talk about "closeness",
convergence and limits.  A metric space is a special, somewhat less
abstract, type of topological space.
==============================================================================

From: George Jones <gjones@uvi.edu>
Subject: Re: Metric Space - what's it about?
Date: Thu, 30 Nov 2000 09:31:04 -0400
Newsgroups: sci.math

In article <904l5r$e3v$1@slb6.atl.mindspring.net>, Kelly McCauley wrote:

> I'm looking at taking a graduate level class - "Metric spaces with
> applications".  I have not come across this math object before and I'm
> wondering what the topic will be like.  I know the definition and such, but
> I'm looking for some hint or advice on what to expect.
>
> I've recently completed a linear algebra course (vector spaces, inner
> products, diagonalizaion, eigenvalues, etc.) and it looks like this is a
> continuation of that sort of topic.

The study of metric spaces is often used as a springboard to the
study of point-set topology.

A metric space is a set X on which there is defined an "intuitive"
notion of distance d between any two points in the set. Intuitive
in the sense

1) for any points x and y in X, d(x,y) is a real number, i.e., the
   distance between x and y

2) if x and y are the same point, then d(x,y) = 0

3) the distance from x to y is the same as the distance from
   y to x, i.e., d(x,y) = d(y,x)

4) the distance traveled going directly from x to y is always
   less than or equal to the distance traveled going from x to
   y, but stopping at z first, i.e., d(x,y) <= d(x,z) + d(z,y)

The inner product spaces that you studied in linear algebra are
all examples of metric spaces. If V is a vector space with inner
product < , >, then d(v1,v2) = (<v1-v2,v1-v2>)^(1/2). However, not
all metric spaces are inner product, or even vector spaces.

Another simple but important example is the set of all real numbers
R, with d(a,b) = |a-b|. This example suggests that many concepts
defined for R can be generalized to metric spaces by replacing
absolute value bars by d's.

For example, f : R -> R is continuous at a if for every epsilon E>0
there is a delta D>0 such that |x-a| < D => |f(x) - f(a)| < E. This
generalizes to metric spaces. If X and Y are metric spaces with
distance functions d and d' respectively, then f : X -> Y is
continuous at x if for every epsilon E>0 there is a delta D>0 such
that d(x,y) < D => d'(f(x),f(a)) < E.

So, a graduate course in metric spaces might (or might not) build
on ideas (like convergence and completeness) seen in an
undergraduate real analysis course.

The idea of opens sets also is easily generalized from R or R^n to
metric spaces. This allows one to talk about some topological ideas,
like compactness, in the context of metric spaces.

These things might be covered in the course; it would be easier to be
sure if I knew the course catalog description or text (if any).
"Applications" could mean many different things, from applications
in physics and engineering, to applications in pure mathematics.

Regards,
George