The ell p spaces

Stub grade: A*

This page is a stub

This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:

Important page

Definition

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

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

Justification for $+\infty$ being included

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

• $$\lim_{p\rightarrow +\infty}(\Vert(x_1,\ldots,x_n)\Vert_p)\eq\Vert(x_1,\ldots,x_n)\Vert_\infty$$

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