Bs space

In the mathematical field of functional analysis, the space bs consists of all infinite sequences (x_i) 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 (x_i) 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 ℓ^∞ 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.

References

• Dunford, N.; Schwartz, J.T. (1958), Linear operators, Part I, Wiley-Interscience.