p-norm on R^n

Example

Let $n \ge 1$ be an integer and let $p \ge 1$ be a real number. For $x = (x_1, \ldots, x_n) \in \mathbb{R}^n$, let $$\|x\|_p = \left( \sum_{i=1}^{n} |x_i|^p \right)^{1/p}, \qquad \|x\|_\infty = \max_{1 \le i \le n} |x_i|.$$ Then $\|\cdot\|_p$ and $\|\cdot\|_\infty$ are norms on $\mathbb{R}^n$. They are $0$ only at $x = 0$, they satisfy $\|\lambda x\| = |\lambda| \, \|x\|$ for real $\lambda$, and they satisfy the triangle inequality $\|x + y\| \le \|x\| + \|y\|$.

Remarks

The triangle inequality for $\|\cdot\|_p$ is the Minkowski inequality. The metric $\|x - y\|_p$ is the p-metric. For $p = 2$, the norm is the Euclidean norm. For each $x$, $\|x\|_p \to \|x\|_\infty$ as $p \to \infty$. All these norms are equivalent, and $\mathbb{R}^n$ is a Banach space with each of them.

Explanations

L1 human

The p-norms give different ways to measure the length of a vector from its coordinates. For $p = 1$, you add the absolute values. For $p = 2$, you get the usual length. For $p = \infty$, you take the largest coordinate.