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 ℓ∞ via the mapping
Furthermore, the space of convergent sequences c is the image of cs under T.