# 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:

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

- Definition:Hilbert Sequence Space
- Definition:Lebesgue Space
- $p$-Sequence Space is Lebesgue Space
- Definition:Space of Bounded Sequences

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

## Sources

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- 2005: René L. Schilling:
*Measures, Integrals and Martingales*... (previous) ... (next): $12.12$ - 2017: Amol Sasane:
*A Friendly Approach to Functional Analysis*... (previous) ... (next): Chapter $1.1$: Normed and Banach spaces. Vector Spaces