Definition:P-Norm/Complex

Definition

Let $p \ge 1$ be a real number.

Let $\C$ denote the complex plane.

Let ${\ell^p}_\C$ denote the complex $p$-sequence space:

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

Let $\mathbf z = \sequence {z_n} \in {\ell^p}_\C$ be a sequence in ${\ell^p}_\C$.

Then the $p$-norm of $\mathbf z$ is defined as:

$\ds \norm {\mathbf z}_p = \paren {\sum_{n \mathop = 0}^\infty \cmod {z_n}^p}^{1 / p}$