‘
p
IS COMPLETE
Let
1
≤
p
≤ ∞
, and recall the definition of the metric space
‘
p
:
For
1
≤
p <
∞
,
‘
p
=
(
sequences
a
= (
a
n
)
∞
n
=1
in
R
such that
∞
X
n
=1

a
n

p
<
∞
)
;
whereas
‘
∞
consists of all those sequences
a
= (
a
n
)
∞
n
=1
such that
sup
n
∈
N

a
n

<
∞
. We
defined the
p
norm as the function
k · k
p
:
‘
p
→
[0
,
∞
)
, given by
k
a
k
p
=
∞
X
n
=1

a
n

p
!
1
/p
,
for
1
≤
p <
∞
,
and
k
a
k
∞
= sup
n
∈
N

a
n

. In class, we showed that the function
d
p
:
‘
p
×
‘
p
→
[0
,
∞
)
given
by
d
p
(
a
,
b
) =
k
a

b
k
p
is actually a metric. We now proceed to show that
(
‘
p
, d
p
)
is a
complete
metric space for
1
≤
p
≤ ∞
. For convenience, we will work with the case
p <
∞
,
as the case
p
=
∞
requires slightly different language (although the same ideas apply).
Suppose that
a
1
,
a
2
,
a
3
, . . .
is a Cauchy sequence in
‘
p
.
Note, each term
a
k
in the se
quence is a point in
‘
p
, and so is itself a sequence:
a
k
= (
a
k
1
, a
k
2
, a
k
3
, . . .
)
.
Now, to say that
(
a
k
)
∞
k
=1
is a Cauchy sequence in
‘
p
is precisely to say that
∀
>
0
∃
K
∈
N
s.t.
∀
k, m
≥
K,
k
a
k

a
m
k
p
<
.
That is, for given
>
0
and sufficiently large
k, m
, we have
∞
X
n
=1

a
k
n

a
m
n

p
=
k
a
k

a
m
k
p
p
<
p
.
Now, the above series has all nonnegative terms, and hence is an upper bound for any
fixed
term in the series. That is to say, for fixed
n
0
∈
N
,

a
k
n
0

a
m
n
0
 ≤
∞
X
n
=1

a
k
n

a
m
n

p
<
p
,
and so we see that the sequence
(
a
k
n
0
)
∞
k
=1
is a Cauchy sequence in
R
. But we know that
R
is a complete metric space, and thus there is a limit
a
n
0
∈
R
to this sequence. This holds
for each
n
0
∈
N
. The following diagram illustrates what’s going on.
a
1
=
a
1
1
a
1
2
a
1
3
a
1
4
· · ·
a
2
=
a
2
1
a
2
2
a
2
3
a
2
4
· · ·
a
3
=
a
3
1
a
3
2
a
3
3
a
3
4
· · ·
a
4
=
a
4
1
a
4
2
a
4
3
a
4
4
· · ·
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
↓
↓
↓
↓
a
1
a
2
a
3
a
4
· · ·
So, we have shown that, in this
‘
p
Cauchy sequence of horizontal sequences, each
vertical
sequence actually converges. Hence, there is a sequence
a
= (
a
1
, a
2
, a
3
, a
4
, . . .
)
to which
“
a
k
converges” in a vague sense. The sense is the “pointwise convergence” along vertical
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 Fall '10
 Prof.KatrinWehrheim
 Metric space, Cauchy sequence, n=1, an p

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