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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 = 0 ∞ x i + 1 y i . {\displaystyle x_{0}\lim _{n\to \infty }y_{n}+\sum _{i=0}^{\infty }x_{i+1}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}.} 
- Sequence space – Vector space of infinite sequences
- Dunford, N.; Schwartz, J.T. (1958), Linear operators, Part I, Wiley-Interscience.