<s>
In	O
functional	B-Application
analysis	I-Application
and	O
related	O
areas	O
of	O
mathematics	O
,	O
a	O
sequence	B-Algorithm
space	I-Algorithm
is	O
a	O
vector	O
space	O
whose	O
elements	O
are	O
infinite	O
sequences	O
of	O
real	O
or	O
complex	O
numbers	O
.	O
</s>
<s>
Equivalently	O
,	O
it	O
is	O
a	O
function	B-Algorithm
space	I-Algorithm
whose	O
elements	O
are	O
functions	O
from	O
the	O
natural	O
numbers	O
to	O
the	O
field	O
K	O
of	O
real	O
or	O
complex	O
numbers	O
.	O
</s>
<s>
All	O
sequence	B-Algorithm
spaces	I-Algorithm
are	O
linear	O
subspaces	O
of	O
this	O
space	O
.	O
</s>
<s>
Sequence	B-Algorithm
spaces	I-Algorithm
are	O
typically	O
equipped	O
with	O
a	O
norm	O
,	O
or	O
at	O
least	O
the	O
structure	O
of	O
a	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
.	O
</s>
<s>
The	O
most	O
important	O
sequence	B-Algorithm
spaces	I-Algorithm
in	O
analysis	O
are	O
the	O
spaces	O
,	O
consisting	O
of	O
the	O
-power	O
summable	O
sequences	O
,	O
with	O
the	O
p-norm	O
.	O
</s>
<s>
Other	O
important	O
classes	O
of	O
sequences	O
like	O
convergent	B-Algorithm
sequences	I-Algorithm
or	O
null	O
sequences	O
form	O
sequence	B-Algorithm
spaces	I-Algorithm
,	O
respectively	O
denoted	O
c	O
and	O
c0	O
,	O
with	O
the	O
sup	O
norm	O
.	O
</s>
<s>
Any	O
sequence	B-Algorithm
space	I-Algorithm
can	O
also	O
be	O
equipped	O
with	O
the	O
topology	B-Architecture
of	O
pointwise	B-Algorithm
convergence	I-Algorithm
,	O
under	O
which	O
it	O
becomes	O
a	O
special	O
kind	O
of	O
Fréchet	B-Algorithm
space	I-Algorithm
called	O
FK-space	B-Algorithm
.	O
</s>
<s>
As	O
a	O
topological	O
space	O
,	O
is	O
naturally	O
endowed	O
with	O
the	O
product	O
topology	B-Architecture
.	O
</s>
<s>
Under	O
this	O
topology	B-Architecture
,	O
is	O
Fréchet	B-Algorithm
,	O
meaning	O
that	O
it	O
is	O
a	O
complete	B-Algorithm
,	O
metrizable	B-Algorithm
,	O
locally	O
convex	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
(	O
TVS	O
)	O
.	O
</s>
<s>
However	O
,	O
this	O
topology	B-Architecture
is	O
rather	O
pathological	O
:	O
there	O
are	O
no	O
continuous	O
norms	O
on	O
(	O
and	O
thus	O
the	O
product	O
topology	B-Architecture
cannot	O
be	O
defined	O
by	O
any	O
norm	O
)	O
.	O
</s>
<s>
Among	O
Fréchet	B-Algorithm
spaces	I-Algorithm
,	O
is	O
minimal	O
in	O
having	O
no	O
continuous	O
norms	O
:	O
</s>
<s>
But	O
the	O
product	O
topology	B-Architecture
is	O
also	O
unavoidable	O
:	O
does	O
not	O
admit	O
a	O
strictly	O
coarser	O
Hausdorff	O
,	O
locally	B-Algorithm
convex	I-Algorithm
topology	I-Algorithm
.	O
</s>
<s>
For	O
that	O
reason	O
,	O
the	O
study	O
of	O
sequences	O
begins	O
by	O
finding	O
a	O
strict	O
linear	O
subspace	O
of	O
interest	O
,	O
and	O
endowing	O
it	O
with	O
a	O
topology	B-Architecture
different	O
from	O
the	O
subspace	O
topology	B-Architecture
.	O
</s>
<s>
defines	O
a	O
norm	O
on	O
In	O
fact	O
,	O
is	O
a	O
complete	B-Algorithm
metric	O
space	O
with	O
respect	O
to	O
this	O
norm	O
,	O
and	O
therefore	O
is	O
a	O
Banach	O
space	O
.	O
</s>
<s>
The	O
set	O
of	O
all	O
convergent	B-Algorithm
sequences	I-Algorithm
is	O
a	O
vector	O
subspace	O
of	O
called	O
the	O
.	O
</s>
<s>
Since	O
every	O
convergent	B-Algorithm
sequence	I-Algorithm
is	O
bounded	O
,	O
is	O
a	O
linear	O
subspace	O
of	O
Moreover	O
,	O
this	O
sequence	B-Algorithm
space	I-Algorithm
is	O
a	O
closed	O
subspace	O
of	O
with	O
respect	O
to	O
the	O
supremum	O
norm	O
,	O
and	O
so	O
it	O
is	O
a	O
Banach	O
space	O
with	O
respect	O
to	O
this	O
norm	O
.	O
</s>
<s>
As	O
a	O
vector	O
space	O
,	O
is	O
equal	O
to	O
,	O
but	O
has	O
a	O
different	O
topology	B-Architecture
.	O
</s>
<s>
This	O
family	O
of	O
inclusions	O
gives	O
a	O
final	O
topology	B-Architecture
,	O
defined	O
to	O
be	O
the	O
finest	O
topology	B-Architecture
on	O
such	O
that	O
all	O
the	O
inclusions	O
are	O
continuous	O
(	O
an	O
example	O
of	O
a	O
coherent	O
topology	B-Architecture
)	O
.	O
</s>
<s>
With	O
this	O
topology	B-Architecture
,	O
becomes	O
a	O
complete	B-Algorithm
,	O
Hausdorff	O
,	O
locally	B-Algorithm
convex	I-Algorithm
,	O
sequential	O
,	O
topological	B-Architecture
vector	I-Architecture
space	I-Architecture
that	O
is	O
Fréchet	B-Algorithm
–	O
Urysohn	O
.	O
</s>
<s>
The	O
topology	B-Architecture
is	O
also	O
strictly	O
finer	O
than	O
the	O
subspace	O
topology	B-Architecture
induced	O
on	O
by	O
.	O
</s>
<s>
Convergence	O
in	O
has	O
a	O
natural	O
description	O
:	O
if	O
and	O
is	O
a	O
sequence	O
in	O
then	O
in	O
if	O
and	O
only	O
is	O
eventually	O
contained	O
in	O
a	O
single	O
image	O
and	O
under	O
the	O
natural	O
topology	B-Architecture
of	O
that	O
image	O
.	O
</s>
<s>
This	O
is	O
facilitated	O
by	O
the	O
fact	O
that	O
the	O
subspace	O
topology	B-Architecture
on	O
,	O
the	O
quotient	O
topology	B-Architecture
from	O
the	O
map	O
,	O
and	O
the	O
Euclidean	O
topology	B-Architecture
on	O
all	O
coincide	O
.	O
</s>
<s>
This	O
shows	O
is	O
an	O
LB-space	B-Algorithm
.	O
</s>
<s>
This	O
set	O
is	O
dense	O
in	O
many	O
sequence	B-Algorithm
spaces	I-Algorithm
.	O
</s>
<s>
In	O
fact	O
,	O
by	O
Pitt	O
's	O
theorem	O
,	O
every	O
bounded	O
linear	B-Architecture
operator	I-Architecture
from	O
to	O
is	O
compact	O
when	O
.	O
</s>
<s>
If	O
1	O
<	O
p	O
<	O
∞	O
,	O
then	O
the	O
(	O
continuous	O
)	O
dual	O
space	O
of	O
ℓp	O
is	O
isometrically	O
isomorphic	O
to	O
ℓq	O
,	O
where	O
q	O
is	O
the	O
Hölder	B-Algorithm
conjugate	I-Algorithm
of	O
p	O
:	O
1/p	O
+	O
1/q	O
=	O
1	O
.	O
</s>
<s>
The	O
space	O
ℓ1	O
has	O
the	O
Schur	B-Algorithm
property	I-Algorithm
:	O
In	O
ℓ1	O
,	O
any	O
sequence	O
that	O
is	O
weakly	B-Algorithm
convergent	I-Algorithm
is	O
also	O
strongly	B-Algorithm
convergent	I-Algorithm
.	O
</s>
<s>
However	O
,	O
since	O
the	O
weak	O
topology	B-Architecture
on	O
infinite-dimensional	O
spaces	O
is	O
strictly	O
weaker	O
than	O
the	O
strong	B-Algorithm
topology	I-Algorithm
,	O
there	O
are	O
nets	O
in	O
ℓ1	O
that	O
are	O
weak	O
convergent	O
but	O
not	O
strong	O
convergent	O
.	O
</s>
