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[parent] $\ell^p(X)$ space (Definition)

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

$\displaystyle \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

$\displaystyle \sum_{x \in X} \vert f(x)\vert^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).

Properties

  • By the corresponding property on $ L^p$-spaces, the space $ \ell^p(X)$ is a Banach space and its norm amounts to
    $\displaystyle \Vert f\Vert _p = \left ( \sum_{x \in X} \vert f(x)\vert^p \right )^{\frac{1}{p}} $

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

\begin{displaymath} \delta_{x_0}(x) := \begin{cases} 1, & $if$\;\; 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$.



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See Also: $\ell^p$, classification of Hilbert spaces

Other names:  $\ell^p(X)$, $\ell^p(X)$-space
Also defines:  $\ell^2(X)$, $\ell^2(X)$ space, $\ell^p(X)$ is nonseparable iff $X$ is uncountable, orthonormal basis of $\ell^2(X)$ have the cardinality of $X$

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Cross-references: isometrically isomorphic, all orthonormal bases have the same cardinality, cardinality, orthonormal basis, Schauder basis, separable, inner product, Hilbert space, norm, Banach space, countable, finite, sum, functions, subsets, counting measure, real number
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This is version 7 of $\ell^p(X)$ space, born on 2008-03-21, modified 2008-03-23.
Object id is 10428, canonical name is EllpXSpace.
Accessed 377 times total.

Classification:
AMS MSC: 28B15 (Measure and integration :: Set functions, measures and integrals with values in abstract spaces :: Set functions, measures and integrals with values in ordered spaces)
 46B26 (Functional analysis :: Normed linear spaces and Banach spaces; Banach lattices :: Nonseparable Banach spaces)
 46E30 (Functional analysis :: Linear function spaces and their duals :: Spaces of measurable functions ($L^p$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant)
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