# Definition:P-Sequence Space

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## Definition

Let $p \in \R$, $p \ge 1$.

The **$p$-sequence space**, denoted $\ell^p$ or $\ell^p \left({\N}\right)$, is defined as:

- $\displaystyle \ell^p := \left\{{\left\langle{z_n}\right\rangle_{n \in \N} \in \C^\N: \sum_{n \mathop = 0}^\infty \left\vert{z_n}\right\vert^p < \infty}\right\}$

As such, $\ell^p$ is a subspace of $\C^\N$, the space of all complex sequences.

## Also defined as

Authors coming from the direction of measure theory often define $\ell^p$ as consisting of real sequences.

To explicate the base field, $\ell^p_\R$ and $\ell^p_\C$ may be used.

## Also known as

Some authors call **sequence spaces** Lebesgue spaces, 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

## Sources

- 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