c space
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In the mathematical field of functional analysis, the space denoted by c is the vector space of all convergent sequences [math]\displaystyle{ \left(x_n\right) }[/math] of real numbers or complex numbers. When equipped with the uniform norm: [math]\displaystyle{ \|x\|_\infty = \sup_n |x_n| }[/math] the space [math]\displaystyle{ c }[/math] becomes a Banach space. It is a closed linear subspace of the space of bounded sequences, [math]\displaystyle{ \ell^\infty }[/math], and contains as a closed subspace the Banach space [math]\displaystyle{ c_0 }[/math] of sequences converging to zero. The dual of [math]\displaystyle{ c }[/math] is isometrically isomorphic to [math]\displaystyle{ \ell^1, }[/math] as is that of [math]\displaystyle{ c_0. }[/math] In particular, neither [math]\displaystyle{ c }[/math] nor [math]\displaystyle{ c_0 }[/math] is reflexive.
In the first case, the isomorphism of [math]\displaystyle{ \ell^1 }[/math] with [math]\displaystyle{ c^* }[/math] is given as follows. If [math]\displaystyle{ \left(x_0, x_1, \ldots\right) \in \ell^1, }[/math] then the pairing with an element [math]\displaystyle{ \left(y_0, y_1, \ldots\right) }[/math] in [math]\displaystyle{ c }[/math] is given by [math]\displaystyle{ x_0\lim_{n\to\infty} y_n + \sum_{i=1}^\infty x_i y_i. }[/math]
This is the Riesz representation theorem on the ordinal [math]\displaystyle{ \omega. }[/math]
For [math]\displaystyle{ c_0, }[/math] the pairing between [math]\displaystyle{ \left(x_i\right) }[/math] in [math]\displaystyle{ \ell^1 }[/math] and [math]\displaystyle{ \left(y_i\right) }[/math] in [math]\displaystyle{ c_0 }[/math] is given by [math]\displaystyle{ \sum_{i=0}^\infty x_iy_i. }[/math]
See also
- Sequence space – Vector space of infinite sequences
References
- Dunford, N.; Schwartz, J.T. (1958), Linear operators, Part I, Wiley-Interscience.
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