C space

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 subspace of , and contains as a closed subspace the Banach space "c"0 of sequences converging to zero. The dual of "c" is isometrically isomorphic to ℓ1, as is that of "c"0. In particular, neither "c" nor "c"0 is reflexive.

In the first case, the isomorphism of ℓ1 with "c"* is given as follows. If ("x"0,"x"1,...) ∈ ℓ1, then the pairing with an element ("y"1,"y"2,...) in "c"* is given by

:x_0lim_{n oinfty} 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 ℓ1 and ("y"i) in "c"0 is given by

:sum_{i=0}^infty x_iy_i.

References

*.


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