c space

In the mathematical field of functional analysis, the space denoted by c is the vector space of all convergent sequences $\left(x_{n}\right)$ of real numbers or complex numbers. When equipped with the uniform norm: $$\|x\|_{\infty }=\sup _{n}|x_{n}|$$ the space $c$ becomes a Banach space. It is a closed linear subspace of the space of bounded sequences, $\ell ^{\infty }$, and contains as a closed subspace the Banach space $c_{0}$ of sequences converging to zero. The dual of $c$ is isometrically isomorphic to $\ell ^{1},$ as is that of $c_{0}.$ In particular, neither $c$ nor $c_{0}$ is reflexive.

In the first case, the isomorphism of $\ell ^{1}$ with $c^{*}$ is given as follows. If $\left(x_{0},x_{1},\ldots \right)\in \ell ^{1},$ then the pairing with an element $\left(y_{0},y_{1},\ldots \right)$ in $c$ is given by $$x_{0}\lim _{n\to \infty }y_{n}+\sum _{i=0}^{\infty }x_{i+1}y_{i}.$$

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

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

See also

• Sequence space – Vector space of infinite sequences

References

• Dunford, N.; Schwartz, J.T. (1958), Linear operators, Part I, Wiley-Interscience.