# Definition:P-Sequence Space/Complex

## 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.