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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.

## Welcome to The Bridge

Mathematics: is it the fabric of MEST?
This is my voyage
My continuous mission
To uncover hidden structures
To create new theorems and proofs
To boldly go where no man has gone before

(Raumpatrouille – Die phantastischen Abenteuer des Raumschiffes Orion, colloquially aka Raumpatrouille Orion was the first German science fiction television series. Its seven episodes were broadcast by ARD beginning September 17, 1966. The series has since acquired cult status in Germany. Broadcast six years before Star Trek first aired in West Germany (in 1972), it became a huge success.)