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Thursday, April 28, 2011

A surprising vector norm

A vector norm of an n-dimensional vector $\mathbf{x}$ denoted by $\parallel\mathbf{x}\parallel$, is a real-valued function of $\mathbf{x}$ such that:
-$\parallel\mathbf{x}\parallel > 0$
-$\parallel k\mathbf{x}\parallel = k \parallel\mathbf{x}\parallel$
-$\parallel \mathbf{x+y}\parallel \le \parallel\mathbf{x}\parallel+ \parallel\mathbf{y}\parallel$

It can simply be verified that the Euclidean length is a vector norm:
$$\parallel\mathbf{x}\parallel = \sqrt{x_1^2 + x_2^2 + \cdots + x_n^x} $$

There is an entire class of norms called the $l_p$-norms:
$$\parallel\mathbf{x}\parallel_p = (x_1^p + x_2^p + \cdots + x_n^p)^{\frac{1}{p}}, $$
of which the Euclidean length or Euclidean norm is a member for $p=2$.

A very interesting ( and surprising ) norm is the $l_\infty$-norm:
$$\parallel\mathbf{x}\parallel_{\infty} = \max{(|x_1|, |x_2|, \cdots, |x_n|)},$$
this behavior can be explained from the fact that the higher the $p$, the more $l_p$ is dominated by the element of the largest magnitude.

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