The ell p spaces

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Definition

Let [ilmath]p\in[1,+\infty]:\eq\{x\in\overline{\mathbb{R} }\ \vert 1\le x\} [/ilmath] be given. We define the [ilmath]\ell^p[/ilmath] normed space as follows:

• If [ilmath]p\in\mathbb{R} [/ilmath] then:
  • [math]\ell^p[/math][math]:\eq\left\{(x_n)_{n\in\mathbb{N} }\subseteq\mathbb{C}\ \left\vert\ \sum_{n\eq 1}^\infty \vert x_n\vert^p<+\infty\right\}\right. [/math]
    • With the norm: [math]\Vert\cdot\Vert_p:(x_n)_{n\in\mathbb{N} }\mapsto\sqrt[p]{\sum^\infty_{n\eq 1}\vert x_n\vert^p } [/math] - often written as [math]\Vert\cdot\Vert_p:(x_n)_{n\in\mathbb{N} }\mapsto\left(\sum^\infty_{n\eq 1}\vert x_n\vert^p\right)^{\frac{1}{p} } [/math] because the pth-root rendering is pretty poor.
• If [ilmath]p\eq+\infty[/ilmath] then:
  • [math]\ell^\infty[/math][math]:\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]
    • with the norm: [math]\Vert\cdot\Vert_\infty:(x_n)_{n\in\mathbb{N} }\mapsto\mathop{\text{Sup} }_{n\in\mathbb{N} }(\vert x_n\vert) [/math]

Justification for [ilmath]+\infty[/ilmath] being included

On [ilmath]\mathbb{R}^n[/ilmath] and [ilmath]\mathbb{C}^n[/ilmath] we also have the p-norm, just as a finite sum rather than an infinite one as shown above. It is claimed that:

• [math]\lim_{p\rightarrow +\infty}(\Vert(x_1,\ldots,x_n)\Vert_p)\eq\Vert(x_1,\ldots,x_n)\Vert_\infty[/math]

The same reference also says the proof that these are norms is basically the same.