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.

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p-metric on R^n example
p-norm on R^n example

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Tag	`0KNJWEVSMG`
URI	`m2://0KNJWEVSMG`
Aliases	`lp-norm-on-rn`
Kind	example
Writing	W0 stub
MSC	46B20
Instance of	Banach space
Uses	p-metric on R^n

p-norm on R^n

Example

Remarks

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