Definition:P-Norm/Complex
< Definition:P-Norm(Redirected from Definition:Complex P-Norm)
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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:
- $\ds \norm {\mathbf z}_p = \paren {\sum_{n \mathop = 0}^\infty \cmod {z_n}^p}^{1 / p}$
Also see
- Results about $p$-norms can be found here.