Definition:P-Sequence Space/Real

Definition
Let $p \in \R$ be a real number such that $p \ge 1$.

Let $\R$ denote the real number line.

The $p$-sequence space (in $\R$), denoted ${\ell^p}_\R$, is defined as:


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

where:
 * $\R^\N$ is the set of all sequences in $\R$
 * $\size {x_n}$ denotes the absolute value of $x_n$.

That is, the $p$-sequence space is the set of all sequences in $\R$ such that $\size {x_n}^p$ converges to a limit.

Also see

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