Английская Википедия:Bs space
In the mathematical field of functional analysis, the space bs consists of all infinite sequences (xi) 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
References
Шаблон:Banach spaces Шаблон:Functional analysis