${\mathrm{\ell}}^{p}(X)$ space
0.0.1 Definition of ${\mathrm{\ell}}^{p}(X)$
Let $p$ be a real number such that $$.
Let $X$ be a set and let $\mu $ be the counting measure on $X$, defined on the $\sigma $algebra (http://planetmath.org/SigmaAlgebra) $\U0001d505$ of all subsets of $X$. The ${\mathrm{\ell}}^{p}(X)$ space is a particular of a ${L}^{p}$space (http://planetmath.org/LpSpace), defined as
$${\mathrm{\ell}}^{p}(X):={L}^{p}(X,\U0001d505,\mu )$$ 
Thus, the ${\mathrm{\ell}}^{p}(X)$ space consists of all functions $f:X\u27f6\u2102$ such that
$$ 
Of course, for the above sum to be finite one must necessarily have $f(x)\ne 0$ 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 ${\mathrm{\ell}}^{p}(X)$ is a Banach space^{} and its norm amounts to
$${\parallel f\parallel}_{p}={\left(\sum _{x\in X}{f(x)}^{p}\right)}^{\frac{1}{p}}$$

•
By the corresponding property on ${L}^{2}$spaces (http://planetmath.org/L2SpacesAreHilbertSpaces), the space ${\mathrm{\ell}}^{2}(X)$ is a Hilbert space^{} and its inner product^{} amounts to
$$\u27e8f,g\u27e9=\sum _{x\in X}f(x)\overline{g(x)}$$
0.0.3 Nonseparability of ${\mathrm{\ell}}^{p}(X)$ for uncountable $X$
 The space ${\mathrm{\ell}}^{p}(X)$ is separable if and only if $X$ is a countable set. Moreover, ${\mathrm{\ell}}^{p}(X)$ admits a Schauder basis^{} if and only if $X$ is countable.
$$
A Schauder basis for ${\mathrm{\ell}}^{p}(X)$, when it exists, can be just the set of functions $\{{\delta}_{{x}_{0}}:{x}_{0}\in X\}$ defined by
$${\delta}_{{x}_{0}}(x):=\{\begin{array}{cc}1,\hfill & \text{if}x={x}_{0}\hfill \\ 0\hfill & \text{if}x\ne {x}_{0}\hfill \end{array}$$ 
0.0.4 Orthonormal basis of ${\mathrm{\ell}}^{2}(X)$
The set of functions $\{{\delta}_{{x}_{0}}:{x}_{0}\in X\}$ is an orthonormal basis^{} of ${\mathrm{\ell}}^{2}(X)$. Hence, the dimension^{} (http://planetmath.org/OrthonormalBasis) of ${\mathrm{\ell}}^{2}(X)$ 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 ${\mathrm{\ell}}^{2}(X)$ space, for a suitable set $X$.
Title  ${\mathrm{\ell}}^{p}(X)$ space 

Canonical name  ellpXSpace 
Date of creation  20130322 17:55:59 
Last modified on  20130322 17:55:59 
Owner  asteroid (17536) 
Last modified by  asteroid (17536) 
Numerical id  10 
Author  asteroid (17536) 
Entry type  Definition 
Classification  msc 46E30 
Classification  msc 46B26 
Classification  msc 28B15 
Synonym  ${\mathrm{\ell}}^{p}(X)$ 
Synonym  ${\mathrm{\ell}}^{p}(X)$space 
Related topic  Lp 
Related topic  ClassificationOfHilbertSpaces 
Related topic  RieszFischerTheorem 
Defines  ${\mathrm{\ell}}^{2}(X)$ 
Defines  ${\mathrm{\ell}}^{2}(X)$ space 
Defines  ${\mathrm{\ell}}^{p}(X)$ is nonseparable iff $X$ is uncountable 
Defines  orthonormal basis of ${\mathrm{\ell}}^{2}(X)$ have the cardinality of $X$ 