1 Distance
J MUSCAT
2
1.0.1
Example
On
N
,
Q
,
R
,
C
and
R
N
, one can take the standard Euclidean distance
d
(
x, y
) =

x

y

. Check that the three axioms for a distance are satisfied (make use
of the fact that

a
+
b


a

+

b

). Note that for
R
n
, the Euclidean distance
d
(
x
,
y
) =
∑
i
=
n
i
=1

x
i

y
i

2
.
One can define distances on other more general spaces, e.g. the space of
continuous functions
f
(
x
) of
x
∈
[0
,
1] has a distance defined by
d
(
f, g
) =
max
x
∈
[0
,
1]

f
(
x
)

g
(
x
)

. The space of shapes (roughly speaking, subsets of
R
2
having an area) have a metric
d
(
A, B
) = area of (
A
∪
B

A
∩
B
). In all
these cases we get an idea of which elements are close together by looking at
their distance.
1.0.2
Exercises
1. Show that if
x
1
, . . . , x
n
are
n
points, then
d
(
x
1
, x
n
)
d
(
x
1
, x
2
) +
. . .
+
d
(
x
n

1
, x
n
).
2. Verify that the metric defined on
R
2
does indeed satisfy the metric
axioms.
3. Show that if
d
is a metric, then so are the maps
D
1
(
x, y
) = 2
d
(
x, y
)
and
D
2
(
x, y
) =
d
(
x,y
)
1+
d
(
x,y
)
, but that
d
(
x, y
)
2
need not be a metric. Hence
a metric space can have several metrics.
4. Show that if
X
,
Y
are metric spaces with distances
d
X
and
d
Y
, then
X
×
Y
is also a metric space with distance
D
(
x
1
y
1
,
x
2
y
2
) =
d
X
(
x
1
, x
2
) +
d
Y
(
y
1
, y
2
)
.
(Note that for
R
2
, this metric is not the Euclidean one.)
5. Show further that
˜
D
(
x
1
y
1
,
x
2
y
2
) = max(
d
X
(
x
1
, x
2
)
, d
Y
(
y
1
, y
2
))
is also a metric for
X
×
Y
.
6. Prove that the function
d
(
m, n
) =

1
/m

1
/n

, on the natural numbers,
is a metric.