Bs space

In the mathematical field of functional analysis, the space bs consists of all infinite sequences (xi) of real numbers $$\R$$ or complex numbers $$\Complex$$ such that $$\sup_n \left|\sum_{i=1}^n x_i\right|$$ is finite. The set of such sequences forms a normed space with the vector space operations defined componentwise, and the norm given by $$\|x\|_{bs} = \sup_n \left|\sum_{i=1}^n x_i\right|.$$

Furthermore, with respect to metric induced by this norm, bs is complete: it is a Banach space.

The space of all sequences $$\left(x_i\right)$$ such that the series $$\sum_{i=1}^\infty x_i$$ 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 $$\ell^{\infty}$$ via the mapping $$T(x_1, x_2, \ldots) = (x_1, x_1+x_2, x_1+x_2+x_3, \ldots).$$

Furthermore, the space of convergent sequences c is the image of cs under $$T.$$