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 c₀ of sequences converging to zero. The dual of c is isometrically isomorphic to ℓ¹, as is that of c₀. In particular, neither c nor c₀ is reflexive.

In the first case, the isomorphism of ℓ¹ with c* is given as follows. If (x₀,x₁,...) ∈ ℓ¹, then the pairing with an element (y₁,y₂,...) 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 c₀, the pairing between (x_i) in ℓ¹ and (y_i) in c₀ is given by

$\sum_{i=0}^\infty x_iy_i.$

References

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