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

${\displaystyle \sup _{n}\left|\sum _{i=1}^{n}x_{i}\right|}$

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

${\displaystyle \left\|x\right\|_{bs}=\sup _{n}\left|\sum _{i=1}^{n}x_{i}\right|.}$

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

${\displaystyle \sum _{i=1}^{\infty }x_{i}}$

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

${\displaystyle T(x_{1},x_{2},\dots )=(x_{1},x_{1}+x_{2},x_{1}+x_{2}+x_{3},\dots ).}$

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.