Euclidean norm ($\Vert x{\Vert }_2$)

The Euclidean norm of $x=(x_1,\dots ,x_n)\in {\mathbb{R}}^n$ is

For a real vector $v$, the square of its Euclidean norm is $\Vert v{\Vert }_2^2=v\cdot v={\sum }_j{v}_j^2$. Unlike the norm itself it is a quadratic function, so expanding sums directly exposes diagonal and cross terms. This is useful for the mean-square displacement of an isotropic planar random walk.

The Euclidean norm is self-dual:The upper bound is Cauchy-Schwarz inequality, and equality holds at $v=x/\Vert x{\Vert }_2$ when $x\ne 0$.

The closed Euclidean ball of radius $R$ about $a\in {\mathbb{R}}^n$ is $\{x:\Vert x-a{\Vert }_2\le R\}$; replacing $\le$ by $<$ gives the open ball.

The unit ball of a normed vector space is $\{x:\Vert x\Vert \le 1\}$ under the closed-ball convention.

The closed unit ball of a normed vector space is the set of vectors of norm at most one. In a continuous dual space it is compact in the weak-star topology by the Banach-Alaoglu theorem.

The open unit ball of a normed vector space $X$ is $B_X=\{x\in X:\Vert x\Vert <1\}$.