Definition:P-Sequence Space

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Definition

Let $p \in \R$ be a real number such that $p \ge 1$.

Let $\BB$ be a Banach space.


The $p$-sequence space (in $\BB$), denoted $\ell^p$ or $\map {\ell^p} \N$, is defined as:

$\ds \ell^p := \set {\sequence {s_n}_{n \mathop \in \N} \in \BB^\N: \sum_{n \mathop = 0}^\infty \norm {s_n}^p < \infty}$

where:

$\BB^\N$ is the set of all sequences in $\BB$
$\norm {s_n}$ denotes the norm of $s_n$.


That is, the $p$-sequence space is the set of all sequences in $\BB$ such that $\norm {s_n}^p$ converges to a limit.


This is often presented in expository treatments either on the real number line or the complex plane:


Real Number Line

The $p$-sequence space (in $\R$), denoted ${\ell^p}_\R$, is defined as:

$\ds {\ell^p}_\R := \set {\sequence {x_n}_{n \mathop \in \N} \in \R^\N: \sum_{n \mathop = 0}^\infty \size {x_n}^p < \infty}$


Complex Plane

The $p$-sequence space (in $\C$), denoted ${\ell^p}_\C$, is defined as:

$\ds {\ell^p}_\C := \set {\sequence {z_n}_{n \mathop \in \N} \in \C^\N: \sum_{n \mathop = 0}^\infty \cmod {z_n}^p < \infty}$


Also known as

Some authors call the $p$-sequence space the Lebesgue space, but this term is reserved for a more general object on $\mathsf{Pr} \infty \mathsf{fWiki}$.


Also see

  • Results about $p$-sequence spaces can be found here.


Sources


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