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c space

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In the mathematical field of functional analysis, the space denoted by c is the vector space of all convergent sequences of real numbers or complex numbers. When equipped with the uniform norm: the space becomes a Banach space. It is a closed linear subspace of the space of bounded sequences, , and contains as a closed subspace the Banach space of sequences converging to zero. The dual of is isometrically isomorphic to as is that of In particular, neither nor is reflexive.

In the first case, the isomorphism of with is given as follows. If then the pairing with an element in is given by

This is the Riesz representation theorem on the ordinal .

For the pairing between in and in is given by

See also

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References

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