c space

In the mathematical field of functional analysis, the space denoted by c is the vector space of all convergent sequences (xn) 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 (xi) in ℓ1 and (yi) in c0 is given by

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

References