# ell^p

Let $\mathbb{F}$ be either $\mathbb{R}$ or $\mathbb{C}$, and let $p\in\mathbb{R}$ with $p\geq 1$. We define $\ell^{p}$ to be the set of all sequences $(a_{i})_{i\geq 0}$ in $\mathbb{F}$ such that

 $\sum_{i=0}^{\infty}|a_{i}|^{p}$

converges.

We also define $\ell^{\infty}$ to be the set of all bounded (http://planetmath.org/BoundedInterval) sequences $(a_{i})_{i\geq 0}$ with norm given by

 $\|(a_{i})\|_{\infty}=\operatorname{sup}\{|a_{i}|:i\geq 0\}.$

By defining addition and scalar multiplication pointwise, $\ell^{p}(\mathbb{F})$ and $\ell^{\infty}(\mathbb{F})$ have a natural vector space stucture. That the sum of two elements on $\ell^{p}(\mathbb{F})$ is again an element in $\ell^{p}(\mathbb{F})$ follows from Minkowski inequality (see below). We can make $\ell^{p}$ into a normed vector space, by defining the norm as

 $\|(a_{i})\|_{p}=(\sum_{i=0}^{\infty}|a_{i}|^{p})^{1/p}.$

The normed vector spaces $\ell^{\infty}$ and $\ell^{p}$ for $p\geq 1$ are complete under these norms, making them into Banach spaces. Moreover, $\ell^{2}$ is a Hilbert space under the inner product

 $\langle(a_{i}),(b_{i})\rangle=\sum_{i=0}^{\infty}a_{i}\overline{b_{i}}$

where $\overline{x}$ denotes the complex conjugate of $x$.

For $p>1$ the (continuous) dual space of $\ell^{p}$ is $\ell^{q}$ where $\frac{1}{p}+\frac{1}{q}=1$, and the dual space of $\ell^{1}$ is $\ell^{\infty}$.

## Properties

1. 1.

If $a=(a_{0},a_{1},\ldots)\in\ell^{p}(\mathbb{F})$ for $1\leq p<\infty$, then $\lim_{k\to\infty}a_{k}=0$. (proof. (http://planetmath.org/ThenA_kto0IfSum_k1inftyA_kConverges))

2. 2.

For $1\leq p<\infty$, $\ell^{p}(\mathbb{F})$ is separable, and $\ell^{\infty}(\mathbb{F})$ is not separable.

3. 3.

Minkowski inequality. If $a,b\in\ell^{p}(\mathbb{F})$ where $p\geq 1$, then

 $\|a+b\|_{p}\leq\|a\|_{p}+\|b\|_{p}.$
Title ell^p Ellp 2013-03-22 12:19:03 2013-03-22 12:19:03 rspuzio (6075) rspuzio (6075) 25 rspuzio (6075) Definition msc 46B99 msc 54E50 EllpXSpace $\ell^{\infty}$ $\ell^{2}$