View source for The little-L(C) spaces are complete

{{DISPLAYTITLE:The {{M|\ell^p(\mathbb{C})}} spaces are complete}}
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==Statement==
Let {{M|p\in[1,+\infty]\subseteq}}[[extended real numbers|{{M|\overline{\mathbb{R} } }}]] be given and consider [[little-L^p(C) space|{{M|\ell^p(\mathbb{C})}}]] then we claim{{rFAVIDMH}}:
# For {{M|p\in\mathbb{R}_{\ge 1} }} that {{MM|\ell^p(\mathbb{C}):\eq\left\{(x_n)_{n\in\mathbb{N} }\subseteq\mathbb{C}\ \left\vert\ \sum^\infty_{n\eq 1}\vert x_n\vert^p<+\infty\right\}\right. }} is a [[complete metric space]] with respect to the [[metric induced by the norm]]: {{MM|\Vert(x_n)_{n\in\mathbb{N} }\Vert_p:\eq\left(\sum^\infty_{n\eq 1}\vert x_n\vert^p\right)^\frac{1}{p} }}
# For {{M|p\eq+\infty}}<ref group="Note">Herein just {{M|\infty}} as only one is in the relevant range of {{M|p}}</ref> that {{MM|\ell^{\infty}(\mathbb{C}):\eq\left\{(x_n)_{n\in\mathbb{N} }\subseteq\mathbb{C}\ \left\vert\ \mathop{\text{Sup} }_{n\in\mathbb{N} }(\vert x_n\vert)<+\infty \right\}\right. }} is a complete metric space with respect to the metric induced by the norm: {{MM|\Vert(x_n)_{n\in\mathbb{N} }\Vert_\infty:\eq\mathop{\text{Sup} }_{n\in\mathbb{N} }(\vert x_n\vert)}}
==Proof of claims==
{{Requires proof|grade=A*|msg=Saving work, will come back to}}
==Notes==
<references group="Note"/>
==References==
<references/>
{{Theorem Of|Analysis|Functional Analysis}}