# 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:

- $\displaystyle \left\Vert \mathbf{x} \right\Vert_p = \left({\sum_{n \mathop = 0}^\infty \left\vert x_n \right\vert^p}\right)^{1/p}$

## 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

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