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Definition 2.14
A sequence (
x
n
) in a metric space (
X
,
d
) is called a
Cauchy sequence
if for an arbitrary
ʵ>
0 there exists
N
∈ N
, such that
d
(
x
m
,
x
n
)
<ʵ
whenever
m
,
n>N
.
Definition 2.15
A metric space is
complete
if every Cauchy sequence is convergent.
For example, the space of real numbers with the usual metric is complete.
Example 2.14
Let
X
be a nonempty set and
F
denote the collection of functions
from
X
to the interval [0, 1]. A metric is defined on
F
as follows:
d
(
f
,
g
):
=
sup
x
∈
X
|
f
(
x
)
−
g
(
x
)
|
.
In fact, (
F
,
d
) is a complete metric space. Let (
f
n
) be a Cauchy sequence in
F
. Then
for every
x
∈
X
, the sequence (
f
n
(
x
)) is Cauchy; and since [0, 1] is complete, the
sequence converges to some
a
x
∈
[0, 1]. Let
f
be the function defined by
f
(
x
)
=
a
x
.
Thus (
f
n
) converges to
f
.
Example 2.15
n
Similar to Example
2.14
, it can be seen that Euclidean
n
-space
R
is complete.
Let
Y
be a subset of a metric space (
X
,
d
) and let
ʵ>
0. A finite set of points
Z
={
z
1
,
z
2
,
...
,
z
m
}
is called an
ʵ-net
for
Y
, if for every point
y
∈
Y
there exists an
z
∈
Z
with
d
(
y
,
z
)
<ʵ
.
Definition 2.16
A subset
Y
of a metric space
X
is
totally bounded
if
Y
possesses
an
ʵ
-net for every
ʵ>
0.
Theorem 2.6
Let
(
X
,
d
)
be a complete metric space. Then Y
ↆ
X is compact iff Y
is closed and totally bounded.
n
Definition 2.17 (Convex Set)
A set
X
ↆ R
is
convex
if for every two points
x
,
y
X
the whole segment between
x
and
y
is also contained in
X
. In other words,
for every
p
∈
∈
[0, 1], the point
px
+
(1
−
p
)
y
belongs to
X
. We write
X
for the
convex closure
of
X
, the smallest convex set containing
X
.
Given two
n
-dimensional vectors
x
=
a
1
,
...
,
a
n
=
b
1
,
...
,
b
n
and
y
, we use
=
i
=
1
a
i
·
the usual definition of dot-product
x
b
i
.
A basic result about convex set is the separability of disjoint convex sets by a
hyperplane [
3
].
·
y
n
be convex sets with X
∩
Y
=∅
Theorem 2.7 (Separation Theorem)
Let X
,
Y
ↆ R
.
n
and a number c
Then there is a hyperplane whose normal is h
∈ R
∈ R
such that
for all x
∈
X and y
∈
Y , we have h
·
x
≤
c
≤
h
·
y
or
for all x
∈
X and y
∈
Y , we have h
·
x
≥
c
≥
h
·
y .
If X and Y are closed and at least one of them is bounded, they can be separated
strictly, i.e. in such a way that
for all x
∈
X and y
∈
Y , we have h
·
x<c<h
·
y
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