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 $$\sum_{i=0}^\infty x_iy_i.$$