Euclidean norm

For $x=(x_1,\dots,x_k)\in\mathbb{R}^k$, the Euclidean norm (also called the $\ell^2$-norm) is

This norm encodes geometric length and induces the standard metric on $\mathbb{R}^k$ via $d(x,y)=\|x-y\|_2$. Many analytic and geometric arguments exploit inequalities involving $\|\cdot\|_2$ (e.g., Cauchy–Schwarz).

Examples:

• If $x=(3,4)\in\mathbb{R}^2$, then $\|x\|_2=5$.
• If $x=(1,1,1)\in\mathbb{R}^3$, then $\|x\|_2=\sqrt{3}$.
• $\|0\|_2=0$ where $0=(0,\dots,0)$.