C space

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: $${\displaystyle \|x\|_{\infty }=\sup _{n}|x_{n}|}$$ the space ${\displaystyle c}$ becomes a Banach space. It is a closed linear subspace of the space of bounded sequences, ${\displaystyle \ell ^{\infty }}$, and contains as a closed subspace the Banach space ${\displaystyle c_{0}}$ of sequences converging to zero. The dual of ${\displaystyle c}$ is isometrically isomorphic to ${\displaystyle \ell ^{1},}$ as is that of ${\displaystyle c_{0}.}$ In particular, neither ${\displaystyle c}$ nor ${\displaystyle c_{0}}$ is reflexive.

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