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: $$\Vert x{\Vert }_{\mathrm{\infty }}=\underset{n}{\sup }|x_n|$$ the space $c$ becomes a Banach space. It is a closed linear subspace of the space of bounded sequences, ${\ell }^{\mathrm{\infty }}$, 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 $(x_0,x_1,\dots )\in {\ell }^1,$ then the pairing with an element $(y_0,y_1,\dots )$ in $c$ is given by $$x_0\underset{n\to \mathrm{\infty }}{\lim }{y}_n+\sum _{i=0}^{\mathrm{\infty }}{x}_{i+1}{y}_i.$$

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 $$\sum _{i=0}^{\mathrm{\infty }}{x}_i{y}_i.$$

See also

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References

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