Questions tagged [normed-spaces]

A vector space $E$, generally over the field $\mathbb R$ or $\mathbb C$ with a map $\lVert \cdot\rVert\colon E\to\mathbb R_+$ is a normed space if the three conditions are fulfilled:

1. $\lVert x\rVert =0\Rightarrow x=0$,
2. For all $x\in E$ and for all $\lambda\in\mathbb R$ (or $\mathbb C$), $\lVert\lambda x\rVert=|\lambda |\lVert x\rVert$,
3. For all $x,y\in E$, $\lVert x+y\rVert\leq\lVert x\rVert+\lVert y\rVert$.

For instance, in $\mathbb{R}^n$ each of the following functions is a norm:

1. $\displaystyle\bigl\lVert(x_1,x_2,\ldots,x_n)\rVert_2=\sqrt{x_1^2+x_2^2+\cdots+x_n^2}$;
2. $\displaystyle\bigl\lVert(x_1,x_2,\ldots,x_n)\rVert_1=|x_1|+|x_2|+\cdots+|x_n|$;
3. $\displaystyle\bigl\lVert(x_1,x_2,\ldots,x_n)\rVert_\infty=\max${$|x_1|,|x_2|,\ldots,|x_n|$}.