<s>
In	O
mathematical	O
analysis	O
,	O
a	O
Banach	B-Algorithm
limit	I-Algorithm
is	O
a	O
continuous	O
linear	B-Algorithm
functional	I-Algorithm
defined	O
on	O
the	O
Banach	O
space	O
of	O
all	O
bounded	O
complex-valued	O
sequences	O
such	O
that	O
for	O
all	O
sequences	O
,	O
in	O
,	O
and	O
complex	O
numbers	O
:	O
</s>
<s>
,	O
where	O
is	O
the	O
shift	B-Algorithm
operator	I-Algorithm
defined	O
by	O
(	O
shift-invariance	O
)	O
;	O
</s>
<s>
if	O
is	O
a	O
convergent	B-Algorithm
sequence	I-Algorithm
,	O
then	O
.	O
</s>
<s>
In	O
other	O
words	O
,	O
a	O
Banach	B-Algorithm
limit	I-Algorithm
extends	O
the	O
usual	O
limits	O
,	O
is	O
linear	O
,	O
shift-invariant	O
and	O
positive	O
.	O
</s>
<s>
However	O
,	O
there	O
exist	O
sequences	O
for	O
which	O
the	O
values	O
of	O
two	O
Banach	B-Algorithm
limits	I-Algorithm
do	O
not	O
agree	O
.	O
</s>
<s>
We	O
say	O
that	O
the	O
Banach	B-Algorithm
limit	I-Algorithm
is	O
not	O
uniquely	O
determined	O
in	O
this	O
case	O
.	O
</s>
<s>
As	O
a	O
consequence	O
of	O
the	O
above	O
properties	O
,	O
a	O
real-valued	O
Banach	B-Algorithm
limit	I-Algorithm
also	O
satisfies	O
:	O
</s>
<s>
The	O
existence	O
of	O
Banach	B-Algorithm
limits	I-Algorithm
is	O
usually	O
proved	O
using	O
the	O
Hahn	O
–	O
Banach	O
theorem	O
(	O
analyst	O
's	O
approach	O
)	O
,	O
or	O
using	O
ultrafilters	O
(	O
this	O
approach	O
is	O
more	O
frequent	O
in	O
set-theoretical	O
expositions	O
)	O
.	O
</s>
<s>
There	O
are	O
non-convergent	O
sequences	O
which	O
have	O
a	O
uniquely	O
determined	O
Banach	B-Algorithm
limit	I-Algorithm
.	O
</s>
<s>
Thus	O
,	O
for	O
any	O
Banach	B-Algorithm
limit	I-Algorithm
,	O
this	O
sequence	O
has	O
limit	O
.	O
</s>
<s>
A	O
bounded	O
sequence	O
with	O
the	O
property	O
,	O
that	O
for	O
every	O
Banach	B-Algorithm
limit	I-Algorithm
the	O
value	O
is	O
the	O
same	O
,	O
is	O
called	O
almost	B-Algorithm
convergent	I-Algorithm
.	O
</s>
<s>
Given	O
a	O
convergent	B-Algorithm
sequence	I-Algorithm
in	O
,	O
the	O
ordinary	O
limit	O
of	O
does	O
not	O
arise	O
from	O
an	O
element	O
of	O
,	O
</s>
<s>
The	O
latter	O
means	O
is	O
the	O
continuous	O
dual	O
space	O
(	O
dual	O
Banach	O
space	O
)	O
of	O
,	O
and	O
consequently	O
,	O
induces	O
continuous	O
linear	B-Algorithm
functionals	I-Algorithm
on	O
,	O
but	O
not	O
all	O
.	O
</s>
<s>
Any	O
Banach	B-Algorithm
limit	I-Algorithm
on	O
is	O
an	O
example	O
of	O
an	O
element	O
of	O
the	O
dual	O
Banach	O
space	O
of	O
which	O
is	O
not	O
in	O
.	O
</s>
