Definition:P-Norm

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

Let $\ell^p$ denote the $p$-sequence space.

Let $\mathbf x = \sequence {x_n} \in \ell^p$.

Then the $p$-norm of $\mathbf x$ is defined as:
 * $\displaystyle \norm {\mathbf x}_p = \paren {\sum_{n \mathop = 0}^\infty \size {x_n}^p}^{1/p}$

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 $p$-norms


 * 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 norm axion N3.


 * For $p > 1$, Minkowski's inequality states that the $p$-norm satisfies the norm axiom N3.


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