Willam-Warnke yield criterion: Difference between revisions

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== however ==
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|real]] or [[complex numbers]] such that


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:<math>\sup_n\left|\sum_{i=1}^n x_i\right|</math>
 
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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


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:<math>\left\|x\right\|_{bs} = \sup_n\left|\sum_{i=1}^n x_i\right|.</math>
 
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Furthermore, with respect to [[metric space|metric]] induced by this norm, ''bs'' is [[complete metric space|complete]]: it is a [[Banach space]].


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The space of all sequences (''x''<sub>''i''</sub>) such that the [[series (mathematics)|series]]
 
 
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:<math>\sum_{i=1}^\infty x_i</math>
 
 
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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.
 
 
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The space ''bs'' is [[isometry|isometrically]] [[isomorphism|isomorphic]] to the [[space of bounded sequences]] ℓ<sup></sup> via the mapping
 
 
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:<math>T(x_1,x_2,\dots) = (x_1,x_1+x_2,x_1+x_2+x_3,\dots).</math>  
 
 
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Furthermore, the [[space of convergent sequences]] [[c space|''c'']] is the image of ''cs'' under ''T''.
 
 
</ul>
==References==
* {{citation|first1=N.|last1=Dunford|first2=J.T.|last2=Schwartz|title=Linear operators, Part I|publisher=Wiley-Interscience|year=1958}}.
 
[[Category:Banach spaces]]
[[Category:Functional analysis]]
{{mathanalysis-stub}}

Revision as of 21:59, 7 March 2013

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

supn|i=1nxi|

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

xbs=supn|i=1nxi|.

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

i=1xi

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

T(x1,x2,)=(x1,x1+x2,x1+x2+x3,).

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

References

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