# 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:

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

- 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