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Space of bounded sequences
In the mathematical field of functional analysis , the space denoted by c is the vector space of all convergent sequences
(
x
n
)
{\displaystyle \left(x_{n}\right)}
of real numbers or complex numbers . When equipped with the uniform norm :
‖
x
‖
∞
=
sup
n
|
x
n
|
{\displaystyle \|x\|_{\infty }=\sup _{n}|x_{n}|}
the space
c
{\displaystyle c}
becomes a
Banach space . It is a
closed linear subspace of the
space of bounded sequences ,
ℓ
∞
{\displaystyle \ell ^{\infty }}
, and contains as a closed subspace the Banach space
c
0
{\displaystyle c_{0}}
of sequences converging to zero. The
dual of
c
{\displaystyle c}
is isometrically isomorphic to
ℓ
1
,
{\displaystyle \ell ^{1},}
as is that of
c
0
.
{\displaystyle c_{0}.}
In particular, neither
c
{\displaystyle c}
nor
c
0
{\displaystyle c_{0}}
is
reflexive .
In the first case, the isomorphism of
ℓ
1
{\displaystyle \ell ^{1}}
with
c
∗
{\displaystyle c^{*}}
is given as follows. If
(
x
0
,
x
1
,
…
)
∈
ℓ
1
,
{\displaystyle \left(x_{0},x_{1},\ldots \right)\in \ell ^{1},}
then the pairing with an element
(
y
0
,
y
1
,
…
)
{\displaystyle \left(y_{0},y_{1},\ldots \right)}
in
c
{\displaystyle c}
is given by
x
0
lim
n
→
∞
y
n
+
∑
i
=
1
∞
x
i
y
i
.
{\displaystyle x_{0}\lim _{n\to \infty }y_{n}+\sum _{i=1}^{\infty }x_{i}y_{i}.}
This is the Riesz representation theorem on the ordinal
ω
.
{\displaystyle \omega .}
For
c
0
,
{\displaystyle c_{0},}
the pairing between
(
x
i
)
{\displaystyle \left(x_{i}\right)}
in
ℓ
1
{\displaystyle \ell ^{1}}
and
(
y
i
)
{\displaystyle \left(y_{i}\right)}
in
c
0
{\displaystyle c_{0}}
is given by
∑
i
=
0
∞
x
i
y
i
.
{\displaystyle \sum _{i=0}^{\infty }x_{i}y_{i}.}
See also
References
Dunford, N.; Schwartz, J.T. (1958), Linear operators, Part I , Wiley-Interscience .
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