Bs space
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In the mathematical field of functional analysis, the space bs consists of all infinite sequences (xi) of real numbers [math]\displaystyle{ \R }[/math] or complex numbers [math]\displaystyle{ \Complex }[/math] such that [math]\displaystyle{ \sup_n \left|\sum_{i=1}^n x_i\right| }[/math] is finite. The set of such sequences forms a normed space with the vector space operations defined componentwise, and the norm given by [math]\displaystyle{ \|x\|_{bs} = \sup_n \left|\sum_{i=1}^n x_i\right|. }[/math]
Furthermore, with respect to metric induced by this norm, bs is complete: it is a Banach space.
The space of all sequences [math]\displaystyle{ \left(x_i\right) }[/math] such that the series [math]\displaystyle{ \sum_{i=1}^\infty x_i }[/math] is convergent (possibly conditionally) is denoted by cs. This is a closed vector subspace of bs, and so is also a Banach space with the same norm.
The space bs is isometrically isomorphic to the Space of bounded sequences [math]\displaystyle{ \ell^{\infty} }[/math] via the mapping [math]\displaystyle{ T(x_1, x_2, \ldots) = (x_1, x_1+x_2, x_1+x_2+x_3, \ldots). }[/math]
Furthermore, the space of convergent sequences c is the image of cs under [math]\displaystyle{ T. }[/math]
See also
References
- Dunford, N.; Schwartz, J.T. (1958), Linear operators, Part I, Wiley-Interscience.
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