View source for C space

{{Short description|Space of bounded sequences}}
{{lowercase}}{{Noinline|date=December 2023}}{{Context|date=December 2023}}

In the [[Mathematics|mathematical]] field of [[functional analysis]], the space denoted by '''''c''''' is the [[vector space]] of all [[convergent sequence]]s <math>\left(x_n\right)</math> of [[real number]]s or [[complex number]]s.  When equipped with the [[uniform norm]]:
<math display="block">\|x\|_\infty = \sup_n |x_n|</math>
the space <math>c</math> becomes a [[Banach space]].  It is a [[Closed set|closed]] [[linear subspace]] of the [[space of bounded sequences]], [[Lp space|<math>\ell^\infty</math>]], and contains as a closed subspace the Banach space <math>c_0</math> of sequences converging to zero.  The [[Dual space|dual]] of <math>c</math> is isometrically isomorphic to <math>\ell^1,</math> as is that of <math>c_0.</math> In particular, neither <math>c</math> nor <math>c_0</math> is [[Reflexive space|reflexive]].

In the first case, the isomorphism of <math>\ell^1</math> with <math>c^*</math> is given as follows.  If <math>\left(x_0, x_1, \ldots\right) \in \ell^1,</math> then the pairing with an element <math>\left(y_0, y_1, \ldots\right)</math> in <math>c</math> is given by
<math display="block">x_0\lim_{n\to\infty} y_n + \sum_{i=0}^\infty x_{i+1} y_i.</math>

This is the [[Riesz representation theorem]] on the [[Ordinal (mathematics)|ordinal]] <math>\omega</math>.

For <math>c_0,</math> the pairing between <math>\left(x_i\right)</math> in <math>\ell^1</math> and <math>\left(y_i\right)</math> in <math>c_0</math> is given by
<math display="block">\sum_{i=0}^\infty x_iy_i.</math>

==See also==

* {{annotated link|Sequence space}}

==References==

{{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]]
[[Category:Normed spaces]]
[[Category:Norms (mathematics)]]


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