Dextrorotation and levorotation

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:

$\Vert x{\Vert }_{\mathrm{\infty }}={\sup }_n\left|x_n\right|$

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

$x_0\underset{n\to \mathrm{\infty }}{\lim }{y}_n+\sum _{i=1}^{\mathrm{\infty }}{x}_i{y}_i.$

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

$\sum _{i=0}^{\mathrm{\infty }}{x}_i{y}_i.$

References

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