Definition:P-Sequence Space

From ProofWiki
Jump to navigation Jump to search

Definition

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

The $p$-sequence space, denoted $\ell^p$ or $\map {\ell^p} \N$, is defined as:

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

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

Sources