Definition:P-Norm

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

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

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

Then the $p$-norm of $\mathbf{x}$ is defined as:
 * $\displaystyle \left\Vert \mathbf{x} \right\Vert_p = \left({\sum_{n \mathop = 0}^\infty \left\vert x_n \right\vert^p}\right)^{1/p}$

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

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


 * 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.