Graphics Reference
In-Depth Information
(1) d(
p
,
q
) ≥ 0.
(2) d(
p
,
q
) = 0 if and only if
p
=
q
.
(3) (symmetry) d(
p
,
q
) = d(
q
,
p
).
(4) (triangle inequality) d(
p
,
r
) £ d(
p
,
q
) + d(
q
,
r
).
The value d(
p
,
q
) is called the
d-distance from
p
to
q
. A
metric space
is a pair (
X
,d),
where d is a metric on
X
.
R
n
admits a number of different metrics. Let
p
,
q
R
n
.
The
standard Euclidean metric
d: d(
p
,
q
) =|
pq
|
n
Â
(
)
=
The
taxicab metric
d
1
:
d
p
,
p
-
q
1
i
i
i
=
1
(
)
=
{
}
The
max metric
d
:
d
p
,
p
-
q
max
•
i
i
1
££
in
Note.
Throughout this topic, whenever we talk about
R
n
, we shall always assume
that its metric is the Euclidean metric unless it is explicitly stated otherwise. The def-
initions and concepts generalize those in Section 4.2.
Many other spaces have metrics. A large source of metric spaces are vector spaces
with an inner product because the distance function between vectors defined by the
inner product is one (Exercise 5.2.2). This applies in particular to many function
spaces.
5.2.1. Example.
The space C
0
([0,1]) of continuous functions on [0,1] can be made
into a vector space by defining the addition of functions and scalar multiplication in
a pointwise fashion. It is easy to check (Exercise 5.2.3) that one possible inner product
on this space is defined by the formula
1
<>=
() ()
Ú
fg
,
ftgtdt
.
(5.1)
0
This inner product leads to the metric
(
)
12
1
2
()
=
Ú
(
()
-
()
)
dfg
,
ft gt dt
.
(5.2)
2
0
Two other metrics on C
0
([0,1]) unrelated to the inner product are
1
Ú
,
()
=
()
-
()
(5.3)
dfg
ft gtdt
1
0
and
()
=
()
-
()
dfg
,
ft gt
.
max
(5.4)
•
01
££
t
