Definition:P-Sequence Space

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

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


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

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 ProofWiki.

Also see

 * Definition:Hilbert Sequence Space
 * Definition:Lebesgue Space
 * Sequence Space is Lebesgue Space