Definition:P-Sequence Space

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:
 * $\BB^\N$ is the set of all sequences in $\BB$
 * $\norm {s_n}$ denotes the norm of $s_n$.

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:

Also see

 * Definition:Hilbert Sequence Space
 * Definition:Lebesgue Space
 * $p$-Sequence Space is Lebesgue Space
 * Definition:Space of Bounded Sequences