# The [ilmath]\ell^p(\mathbb{C})[/ilmath] spaces are complete

## 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]:

1. For [ilmath]p\in\mathbb{R}_{\ge 1} [/ilmath] that $\ell^p(\mathbb{C}):\eq\left\{(x_n)_{n\in\mathbb{N} }\subseteq\mathbb{C}\ \left\vert\ \sum^\infty_{n\eq 1}\vert x_n\vert^p<+\infty\right\}\right.$ is a complete metric space with respect to the metric induced by the norm: $\Vert(x_n)_{n\in\mathbb{N} }\Vert_p:\eq\left(\sum^\infty_{n\eq 1}\vert x_n\vert^p\right)^\frac{1}{p}$
2. For [ilmath]p\eq+\infty[/ilmath][Note 1] that $\ell^{\infty}(\mathbb{C}):\eq\left\{(x_n)_{n\in\mathbb{N} }\subseteq\mathbb{C}\ \left\vert\ \mathop{\text{Sup} }_{n\in\mathbb{N} }(\vert x_n\vert)<+\infty \right\}\right.$ is a complete metric space with respect to the metric induced by the norm: $\Vert(x_n)_{n\in\mathbb{N} }\Vert_\infty:\eq\mathop{\text{Sup} }_{n\in\mathbb{N} }(\vert x_n\vert)$