# Bs space

In the mathematical field of functional analysis, the space bs consists of all infinite sequences (xi) of real or complex numbers 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

$\left\|x\right\|_{bs}=\sup _{n}\left|\sum _{i=1}^{n}x_{i}\right|.$ Furthermore, with respect to metric induced by this norm, bs is complete: it is a Banach space.

The space of all sequences (xi) 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 via the mapping

$T(x_{1},x_{2},\dots )=(x_{1},x_{1}+x_{2},x_{1}+x_{2}+x_{3},\dots ).$ Furthermore, the space of convergent sequences c is the image of cs under T.

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