Econ 4111
Professor: John Nachbar
11/20/08
Metric Spaces
1
Metric Spaces Basics.
1.1
Metric spaces.
A
metric space
(
X, d
) consists of a set of points,
X
together with a distance function,
or metric,
d
:
X
×
X
→
R
. The interpretation is that
d
(
a, b
) is the distance between
a
and
b
. To qualify as a distance function,
d
must satisfy three properties.
1. For any
a, b
∈
X
,
d
(
a, b
)
≥
0 with
d
(
a, b
) = 0 iff
a
=
b
.
2. For any
a, b
∈
X
,
d
(
a, b
) =
d
(
b, a
).
3. For any
a, b, c
∈
X
,
d
(
a, c
)
≤
d
(
a, b
) +
d
(
b, c
). (The
triangle inequality
.)
The triangle inequality gets its name because, in standard geometry, if the three
points
a
,
b
, and
c
form a triangle then the length of the side from
a
to
c
is less then
the sum of the lengths of the other two sides.
Think of
X
as fundamental while the metric
d
is a kind of overlay, like the
grid on a map, that we add to help with our analysis.
Any
X
has an infinity of
possible metrics. At a minimum, metrics can differ because of units (inches rather
than centimeters). But it is possible for different metrics to give different answers
to questions like, “Is
a
further from
c
than
b
is from
c
?” (I give an example of this
below).
Which metic we choose depends entirely on which is most helpful to us.
For some spaces, notably
X
=
R
N
, there is a default metric that is convenient for
almost any application. For other spaces, such as variants of
X
=
R
∞
, there is no
default metric.
1.2
The space
R
N
.
The most familiar example of a metric space is
R
N
, which is the space of points
of the form (
x
1
, . . . , x
N
), where each
x
n
∈
R
.
For
R
N
, the default metric is the
Euclidean metric
d
E
defined by for all
a, b
∈
R
N
,
d
E
(
a, b
) =
q
(
a

b
)
·
(
a

b
) =
s
X
n
(
a
n

b
n
)
2
.
In
R
,
R
2
, or
R
3
, the Euclidean metric corresponds to the everyday notion of physical
distance. It is easiest to see this if I focus on the special case of measuring distance
to the origin. Given
x
∈
R
N
, define
k
x
k
=
d
E
(
x,
0)
.
1
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Then
k
x
k
, called the
Euclidean norm
of
x
, measures the distance of
x
from the
origin. For example, suppose that
x
= (
x
1
, x
2
)
∈
R
2
. Then, the distance from the
origin to
x
is the length of the hypotenuse of the right triangle with short sides given
by (
x
1
,
0) and (0
, x
2
). By the Pythagorean theorem, the length of this hypotenuse
is
q
x
2
1
+
x
2
2
.
To be careful, I have to verify that
d
E
is
a metric.
Theorem 1.
The Euclidean metric on
R
N
satisfies the three metric properties.
Of the metric properties, only the triangle inequality requires any work. To prove
that, I first establish an important inequality called the CauchySchwartz inequality.
Theorem 2
(CauchySchwartz)
.
For any
a, b
∈
R
N
,

a
·
b
 ≤ k
a
kk
b
k
Proof of Theorem 2.
If
a
·
b
= 0 then the result is immediate. Suppose
a
·
b
6
= 0.
For any
λ
≥
0,
0
≤ k
a

λb
k
2
=
k
a
k
2

2
λ
(
a
·
b
) +
λ
2
k
b
k
2
Since
a
·
b
6
= 0, I can define
λ
=
k
a
k
2
a
·
b
.
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 Fall '08
 JohnNachbar
 Topology, Metric space, Topological space, Cauchy, dmax

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