In the mathematical field of functional analysis, the space denoted by c is the vector space of all convergent sequences ${\displaystyle \left(x_{n}\right)}$ of real numbers or complex numbers. When equipped with the uniform norm:

$${\displaystyle \|x\|_{\infty }=\sup _{n}|x_{n}|}$$

${\displaystyle c}$

${\displaystyle \ell ^{\infty ))$

${\displaystyle c_{0))$

${\displaystyle c}$

${\displaystyle \ell ^{1},}$

${\displaystyle c_{0}.}$

${\displaystyle c}$

${\displaystyle c_{0))$

In the first case, the isomorphism of ${\displaystyle \ell ^{1))$ with ${\displaystyle c^{*))$ is given as follows. If ${\displaystyle \left(x_{0},x_{1},\ldots \right)\in \ell ^{1},}$ then the pairing with an element ${\displaystyle \left(y_{0},y_{1},\ldots \right)}$ in ${\displaystyle c}$ is given by

$${\displaystyle x_{0}\lim _{n\to \infty }y_{n}+\sum _{i=0}^{\infty }x_{i+1}y_{i}.}$$

This is the Riesz representation theorem on the ordinal ${\displaystyle \omega }$.

For ${\displaystyle c_{0},}$ the pairing between ${\displaystyle \left(x_{i}\right)}$ in ${\displaystyle \ell ^{1))$ and ${\displaystyle \left(y_{i}\right)}$ in ${\displaystyle c_{0))$ is given by

$${\displaystyle \sum _{i=0}^{\infty }x_{i}y_{i}.}$$