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# c space

In the mathematical field of functional analysis, the space denoted by * c* is the vector space of all convergent sequences (

*x*

_{n}) 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, ℓ^{∞}, and contains as a closed subspace the Banach space *c*_{0} of sequences converging to zero. The dual of *c* is isometrically isomorphic to ℓ^{1}, as is that of *c*_{0}. In particular, neither *c* nor *c*_{0} is reflexive.

In the first case, the isomorphism of ℓ^{1} with *c** is given as follows. If (*x*_{0},*x*_{1},...) ∈ ℓ^{1}, then the pairing with an element (*y*_{1},*y*_{2},...) in *c* is given by

\( x_0\lim_{n\to\infty} y_n + \sum_{i=1}^\infty x_i y_i. \)

This is the Riesz representation theorem on the ordinal ω.

For *c*_{0}, the pairing between (*x*_{i}) in ℓ^{1} and (*y*_{i}) in *c*_{0} is given by

\( \sum_{i=0}^\infty x_iy_i. \)

References

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

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