Английская Википедия:Bs space

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In the mathematical field of functional analysis, the space bs consists of all infinite sequences (x_i) of real numbers <math>\R</math> or complex numbers <math>\Complex</math> such that <math display="block">\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 display="block">\|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>\left(x_i\right)</math> such that the series <math display="block">\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>\ell^{\infty}</math> via the mapping <math display="block">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>T.</math>

See also

• Шаблон:Annotated link

References

• Шаблон:Citation.

Шаблон:Mathanalysis-stub