C space explained

In the mathematical field of functional analysis, the space denoted by c is the vector space of all convergent sequences

\left(x _{n\right)} of real numbers or complex numbers. When equipped with the uniform norm:$$\backslash |x\backslash |\_\backslash infty\; =\; \backslash sup\_n\; |x\_n|$$the space c becomes a Banach space. It is a closed linear subspace of the space of bounded sequences, \ell ^{infty} , and contains as a closed subspace the Banach space c _{0} of sequences converging to zero. The dual of c is isometrically isomorphic to \ell ^{1,} as is that of c _{0.} In particular, neither c nor c _{0} is reflexive.

In the first case, the isomorphism of

\ell ^{1} with c ^{*} is given as follows. If \left(x _{0,} x _{1,} \ldots\right) \in \ell ^{1,} then the pairing with an element \left(y _{0,} y _{1,} \ldots\right) in c is given by$$x\_0\backslash lim\_\; y\_n\; +\; \backslash sum\_^\backslash infty\; x\_\; y\_i.$$\omega .

For

c _{0,} the pairing between \left(x _{i\right)} in \ell ^{1} and \left(y _{i\right)} in c _{0} is given by$$\backslash sum\_^\backslash infty\; x\_iy\_i.$$

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