# Bs space

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

:$sup_nleft|sum_\left\{i=1\right\}^n x_i ight|$

is finite. The set of such sequences forms a normed space with the vector space operations defined componentwise, and the norm given by

:$left|x ight|_\left\{bs\right\} = sup_nleft|sum_\left\{i=1\right\}^n x_i ight|.$

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

:$sum_\left\{i=1\right\}^infty x_i$

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 ℓ via the mapping

:$T\left(x_1,x_2,dots\right) = \left(x_1,x_1+x_2,x_1+x_2+x_3,dots\right).$

Furthermore, "c" is the image of "cs" under "T".

References

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