Bs space

In the mathematical field of functional analysis, the space bs consists of all infinite sequences (xi) of real numbers R {\displaystyle \mathbb {R} } or complex numbers C {\displaystyle \mathbb {C} } such that

sup n | ∑ i = 1 n x i | {\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

‖ x ‖ b s = sup n | ∑ i = 1 n x i | . {\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 ( x i ) {\displaystyle \left(x_{i}\right)} such that the series

∑ i = 1 ∞ x i {\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 ∞ {\displaystyle \ell ^{\infty }} via the mapping

T ( x 1 , x 2 , … ) = ( x 1 , x 1 + x 2 , x 1 + x 2 + x 3 , … ) . {\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 T . {\displaystyle T.}

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