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 $\mathfrak{B}$ of all subsets of $X$ . The $\ell^p(X)$ space is a particular type of a $L^p$ -space, defined as

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

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).

Properties

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

• By the corresponding property on $L^2$ -spaces, the space $\ell^2(X)$ is a Hilbert space and its inner product amounts to
  

Nonseparability of $\ell^p(X)$ for uncountable $X$

Proposition - 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

$if$ \;\; x=x_0\\ 0 & $if$ \;\; x \neq x_0 \end{cases}\end{displaymath}">

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 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 $X$ .