Definition:P-Norm

Definition
Let $p \ge 1$ be a real number, and let $\mathbf{x} = \langle{x_n}\rangle$ be a member of the Lebesgue space $\ell^p$.

The $p$-norm of $\mathbf{x}$ is defined as:
 * $\displaystyle \left\Vert \mathbf{x} \right\Vert_p = \left({\sum_{n=1}^\infty \left\vert x_n \right\vert^p}\right)^{1/p}$

Remark
The identity $\left\Vert \mathbf{x}^p \right\Vert_q = \left\Vert \mathbf{x} \right\Vert_{pq}^p$ can be used to transform expressions involving $p$-norms.

Also see

 * By the triangle inequality, the $1$-norm satisfies the norm axiom N3. The $1$-norm is also known as the taxicab norm.


 * 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.