Category:Definitions/P-Norms

This category contains definitions related to $p$-norms.
Related results can be found in Category:P-Norms.

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

Let $\BB$ be a Banach space.

Let $\ell^p$ denote the $p$-sequence space in $\BB$:

$\ds \ell^p := \set {\sequence {s_n}_{n \mathop \in \N} \in \BB^\N: \sum_{n \mathop = 0}^\infty \norm {s_n}^p < \infty}$

Let $\mathbf s = \sequence {s_n} \in \ell^p$ be a sequence in $\ell^p$.

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

$\ds \norm {\mathbf s}_p = \paren {\sum_{n \mathop = 0}^\infty \size {s_n}^p}^{1 / p}$