Definition:P-Norm

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

This is often presented in expository treatments either on the real number line or the complex plane:

Also see

 * $p$-Norm is Norm
 * Derivative of $p$-Norm wrt $p$
 * $p$-Norm of Real Sequence is Strictly Decreasing Function of $p$
 * Transformation of $p$-Norm


 * Definition:Euclidean Norm: for $p = 2$, that is, the $2$-norm


 * Definition:Taxicab Norm: for $p = 1$, that is, the $1$-norm


 * By the triangle inequality, the $1$-norm satisfies the.


 * For $p > 1$, Minkowski's Inequality for Sums states that the $p$-norm satisfies the.


 * The $p$-norm is not to be confused with the $p$-adic norm.