# $\ell^{p}(X)$ space

property

## 0.0.1 Definition of $\ell^{p}(X)$

Let $p$ be a real number such that $1\leq p<\infty$.

Let $X$ be a set and let $\mu$ be the counting measure on $X$, defined on the $\sigma$-algebra (http://planetmath.org/SigmaAlgebra) $\mathfrak{B}$ of all subsets of $X$. The $\ell^{p}(X)$ space is a particular of a $L^{p}$-space (http://planetmath.org/LpSpace), defined as

 $\ell^{p}(X):=L^{p}(X,\mathfrak{B},\mu)$

Thus, the $\ell^{p}(X)$ space consists of all functions $f:X\longrightarrow\mathbb{C}$ such that

 $\sum_{x\in X}|f(x)|^{p}<\infty$

Of course, for the above sum to be finite one must necessarily have $f(x)\neq 0$ only for a countable number of $x\in X$ (see this entry (http://planetmath.org/SupportOfIntegrableFunctionWithRespectToCountingMeasureIsCountable)).

## 0.0.2 Properties

• By the corresponding property on $L^{p}$-spaces, the space $\ell^{p}(X)$ is a Banach space and its norm amounts to

 $\|f\|_{p}=\left(\sum_{x\in X}|f(x)|^{p}\right)^{\frac{1}{p}}$
• By the corresponding property on $L^{2}$-spaces (http://planetmath.org/L2SpacesAreHilbertSpaces), the space $\ell^{2}(X)$ is a Hilbert space and its inner product amounts to

 $\langle f,g\rangle=\sum_{x\in X}f(x)\overline{g(x)}$

## 0.0.3 Nonseparability of $\ell^{p}(X)$ for uncountable $X$

- The space $\ell^{p}(X)$ is separable if and only if $X$ is a countable set. Moreover, $\ell^{p}(X)$ admits a Schauder basis if and only if $X$ is countable.

$\,$

A Schauder basis for $\ell^{p}(X)$, when it exists, can be just the set of functions $\{\delta_{x_{0}}:x_{0}\in X\}$ defined by

 $\delta_{x_{0}}(x):=\begin{cases}1,&if\;\;x=x_{0}\\ 0&if\;\;x\neq x_{0}\end{cases}$

## 0.0.4 Orthonormal basis of $\ell^{2}(X)$

The set of functions $\{\delta_{x_{0}}:x_{0}\in X\}$ is an orthonormal basis of $\ell^{2}(X)$. Hence, the dimension (http://planetmath.org/OrthonormalBasis) of $\ell^{2}(X)$ is given by the cardinality of $X$ (as all orthonormal bases have the same cardinality).

It can be shown that all Hilbert spaces are isometrically isomorphic (hence, preserving the inner product) to a $\ell^{2}(X)$ space, for a suitable set $X$.

Title $\ell^{p}(X)$ space ellpXSpace 2013-03-22 17:55:59 2013-03-22 17:55:59 asteroid (17536) asteroid (17536) 10 asteroid (17536) Definition msc 46E30 msc 46B26 msc 28B15 $\ell^{p}(X)$ $\ell^{p}(X)$-space Lp ClassificationOfHilbertSpaces RieszFischerTheorem $\ell^{2}(X)$ $\ell^{2}(X)$ space $\ell^{p}(X)$ is nonseparable iff $X$ is uncountable orthonormal basis of $\ell^{2}(X)$ have the cardinality of $X$