Definition:P-Sequence Space/Complex

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Definition

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

Let $\C$ denote the complex plane.


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

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

where:

$\C^\N$ is the set of all sequences in $\C$
$\cmod {z_n}$ denotes the modulus of $z_n$.


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

Thus ${\ell^p}_\C$ is a subspace of $\C^\N$, the space of all complex sequences.


Also denoted as

The complex $p$-sequence space ${\ell^p}_\C$ is often denoted just as $\ell^p$ when there is no confusion as to what the underlying set is.


Some sources use the form $\map { {\ell^p}_\C} \N$ when it is necessary to bring attention to the fact that the domain of the sequences is the natural numbers.

That is, that the sequences in question are infinite.


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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