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 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 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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