Euclidean Norm

The length function ||x|| = sqrt(x1^2 + ... + xk^2) on R^k

Euclidean Norm

The Euclidean norm on Euclidean space $\mathbb{R}^k$ is the function $\|\cdot\|:\mathbb{R}^k\to\mathbb{R}_{\ge 0}$ defined by

\|x\|=\sqrt{x_1^2+\cdots+x_k^2}\quad\text{for }x=(x_1,\dots,x_k).

It can also be expressed in terms of the standard inner product by $\|x\|=\sqrt{\langle x,x\rangle}$ .

The Euclidean norm is compatible with the absolute value on $\mathbb{R}$ in the sense that for $k=1$ , $\|x\|=|x|$ .

Examples:

In $\mathbb{R}^2$ , $\|(3,4)\|=5$ .
For the standard basis vector $e_i\in\mathbb{R}^k$ (with a $1$ in the $i$ th coordinate and $0$ elsewhere), one has $\|e_i\|=1$ .
The distance between $x,y\in\mathbb{R}^k$ is $\|x-y\|$ .