# Definition:P-Sequence Space/Real

## Definition

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

Let $\R$ denote the real number line.

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

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

where:

- $\R^\N$ is the set of all sequences in $\R$
- $\size {x_n}$ denotes the absolute value of $x_n$.

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

## Also denoted as

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

Some sources use the form $\map { {\ell^p}_\R} \N$ 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