Definition: Vector norm

Informally, a (vector) norm is a function which assigns a length to vectors.

Any sensible measure of length should satisfy the following basic properties: it should be a convex function of its argument (that is, the length of an average of two vectors should be always less than the average of their lengths); it should be positive-definite (always non-negative, and zero only when the argument is the zero vector), and preserve positive scaling (so that multiplying a vector by a positive number scales its norm accordingly).

Formally, a vector norm is a function f:\mathbb{R}^n \rightarrow \mathbb{R} which satisfies the following properties.

Definition of a vector norm
 

1. Positive homogeneity:  for every x \in \mathbb{R}^n, \alpha \ge 0, we have f(\alpha x) = \alpha f(x).

2. Triangle inequality:  for every x, y \in \mathbb{R}^n , we have

    \begin{align*} f(x+y) &\leq f(x)+f(y) \end{align*}

3. Definiteness:  for every x \in \mathbb{R}^n, f(x)=0 implies x=0.

A consequence of the first two conditions is that a norm only assumes non-negative values, and that it is convex.

Popular norms include the so-called l_p-norms, where p=1,2 or p=\infty:

    \begin{align*} \|x\|_p &:= \left(\sum\limits_{i=1}^{n} |x_i|^p\right)^{1/p}, \end{align*}

with the convention that when

    \begin{align*} p = \infty, \quad \|x\|_\infty &= \max_{1\leq i \leq n} |x_i|. \end{align*}

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