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

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

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

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

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


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