PlanetMath: $\ell^p$

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$\ell^p$

(Definition)

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

$\displaystyle \sum_{i=0}^{\infty}\vert a_i\vert^p$

converges.

We also define $\ell^{\infty}$ to be the set of all bounded sequences $(a_i)_{i\geq 0}$ with norm given by

$\displaystyle \Vert (a_i)\Vert_{\infty} = \operatorname{sup}\{ \vert a_i\vert:i\geq 0\}.$

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

$\displaystyle \Vert (a_i)\Vert_p = (\sum_{i=0}^{\infty}\vert a_i\vert^p)^{1/p}.$

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

$\displaystyle \langle (a_i),(b_i)\rangle = \sum_{i=0}^{\infty}a_i \overline{b_i}$

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

For 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

If $a=(a_0,a_1, \ldots ) \in \ell^p(\mathbb{F})$ for $1\le p< \infty$ , then $\lim_{k\to \infty} a_k =0$ . (proof.)
For $1\le p<\infty$ , $\ell^p(\mathbb{F})$ is separable, and $\ell^\infty(\mathbb{F})$ is not separable.
Minkowski inequality. If $a,b\in \ell^p(\mathbb{F})$ where $p\ge 1$ , then
$\displaystyle \Vert a+b \Vert_p \le \Vert a\Vert_p + \Vert b \Vert_p.$

" $\ell^p$ " is owned by rspuzio. [ full author list (3) | owner history (1) ]

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See Also: $\ell^p(X)$ space

Also defines:

$\ell^\infty$ , $\ell^2$

Attachments:: shift operators in $\ell^p$ (Definition) by matte

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Cross-references: separable, dual space, continuous, complex conjugate, inner product, Hilbert space, Banach spaces, complete, normed vector space, Minkowski inequality, sum, vector space, pointwise, multiplication, scalar, addition, norm, converges, sequences
There are 2 references to this entry.

This is version 19 of $\ell^p$ , born on 2002-02-13, modified 2005-05-21.
Object id is 1929, canonical name is Lp.
Accessed 5376 times total.

Classification:

AMS MSC:	46B99 (Functional analysis :: Normed linear spaces and Banach spaces; Banach lattices :: Miscellaneous)
	54E50 (General topology :: Spaces with richer structures :: Complete metric spaces)

Pending Errata and Addenda

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