Difference between revisions of "The little-L(C) spaces are complete"
From Maths
(Created page with "{{DISPLAYTITLE:The {{M|\ell^p(\mathbb{C})}} spaces are complete}} __TOC__ ==Statement== Let {{M|p\in[1,+\infty]\subseteq}}extended real numbers|{{M|\overline{\mathbb{R} } }}...") |
(No difference)
|
Latest revision as of 19:03, 18 March 2017
Contents
Statement
Let [ilmath]p\in[1,+\infty]\subseteq[/ilmath][ilmath]\overline{\mathbb{R} } [/ilmath] be given and consider [ilmath]\ell^p(\mathbb{C})[/ilmath] then we claim[1]:
- For [ilmath]p\in\mathbb{R}_{\ge 1} [/ilmath] that [math]\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. [/math] is a complete metric space with respect to the metric induced by the norm: [math]\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} [/math]
- For [ilmath]p\eq+\infty[/ilmath][Note 1] that [math]\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. [/math] is a complete metric space with respect to the metric induced by the norm: [math]\Vert(x_n)_{n\in\mathbb{N} }\Vert_\infty:\eq\mathop{\text{Sup} }_{n\in\mathbb{N} }(\vert x_n\vert)[/math]
Proof of claims
Grade: A*
This page requires one or more proofs to be filled in, it is on a to-do list for being expanded with them.
Please note that this does not mean the content is unreliable. Unless there are any caveats mentioned below the statement comes from a reliable source. As always, Warnings and limitations will be clearly shown and possibly highlighted if very important (see template:Caution et al).
The message provided is:
The message provided is:
Saving work, will come back to
Notes
- ↑ Herein just [ilmath]\infty[/ilmath] as only one is in the relevant range of [ilmath]p[/ilmath]
References
