Bs space

In the mathematical field of functional analysis, the space bs consists of all infinite sequences (x_i) of real numbers ${\displaystyle \mathbb {R} }$ or complex numbers ${\displaystyle \mathbb {C} }$ 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 $${\displaystyle \|x\|_{bs}=\sup _{n}\left|\sum _{i=1}^{n}x_{i}\right|.}$$

Furthermore, with respect to metric induced by this norm, bs is complete: it is a Banach space.

The space of all sequences ${\displaystyle \left(x_{i}\right)}$ 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 ${\displaystyle \ell ^{\infty }}$ via the mapping $${\displaystyle T(x_{1},x_{2},\ldots )=(x_{1},x_{1}+x_{2},x_{1}+x_{2}+x_{3},\ldots ).}$$

Furthermore, the space of convergent sequences c is the image of cs under ${\displaystyle T.}$