View source for Bs space

{{Short description|Mathematical concept concerning infinite sequences}}
In the [[Mathematics|mathematical]] field of [[functional analysis]], the space '''bs''' consists of all infinite [[Sequence (mathematics)|sequences]] (''x''<sub>''i''</sub>) of [[real numbers]] <math>\R</math> or [[complex numbers]] <math>\Complex</math> such that
<math display="block">\sup_n \left|\sum_{i=1}^n x_i\right|</math>
is finite.  The set of such sequences forms a [[normed space]] with the [[vector space]] operations defined [[Componentwise operation|componentwise]], and the norm given by
<math display="block">\|x\|_{bs} = \sup_n \left|\sum_{i=1}^n x_i\right|.</math>

Furthermore, with respect to [[Metric (mathematics)|metric]] induced by this norm, ''bs'' is [[Complete metric space|complete]]: it is a [[Banach space]].

The space of all sequences <math>\left(x_i\right)</math> such that the [[Series (mathematics)|series]]
<math display="block">\sum_{i=1}^\infty x_i</math>
is [[Convergent series|convergent]] (possibly [[Conditional convergence|conditionally]]) is denoted by ''cs''.  This is a [[Closed set|closed]] [[vector subspace]] of ''bs'', and so is also a Banach space with the same norm.

The space ''bs'' is [[Isometry|isometrically]] [[Isomorphism|isomorphic]] to the [[Space of bounded sequences]] <math>\ell^{\infty}</math> via the mapping
<math display="block">T(x_1, x_2, \ldots) = (x_1, x_1+x_2, x_1+x_2+x_3, \ldots).</math> 

Furthermore, the [[space of convergent sequences]] [[c space|''c'']] is the image of ''cs'' under <math>T.</math>

== See also ==

* {{annotated link|List of Banach spaces}}

== References ==

{{reflist|group=note}}
{{reflist}}

* {{citation|first1=N.|last1=Dunford|first2=J.T.|last2=Schwartz|title=Linear operators, Part I|publisher=Wiley-Interscience|year=1958}}.

{{Banach spaces}}
{{Functional analysis}}

[[Category:Banach spaces]]
[[Category:Functional analysis]]


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