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

property

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) $𝔅$ 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

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

Thus, the ${\mathrm{\ell }}^p(X)$ space consists of all functions $f:X⟶ℂ$ 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

  $$⟨f,g⟩=\sum _{x\in X}f(x)\overline{g(x)}$$

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

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 $X$.