Definition:P-Sequence Space/Complex

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

Let $\C$ denote the complex plane.

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


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

where:
 * $\C^\N$ is the set of all sequences in $\C$
 * $\cmod {z_n}$ denotes the modulus of $z_n$.

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

Thus ${\ell^p}_\C$ is a subspace of $\C^\N$, the space of all complex sequences.

Also see

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