Difference between revisions of "Norm"

Revision as of 16:18, 7 March 2015

A norm on a vector space $(V,F)$ is a function $\|\cdot\|:V\rightarrow\mathbb{R}$ such that:

$\forall x\in V\ \|x\|\ge 0$
$\|x\|=0\iff x=0$
$\forall \lambda\in F, x\in V\ \|\lambda x\|=|\lambda|\|x\|$ where $|\cdot|$ denotes absolute value
$\forall x,y\in V\ \|x+y\|\le\|x\|+\|y\|$ - a form of the triangle inequality

Often parts 1 and 2 are combined into the statement

I don't like this

The Euclidean norm is denoted $\|\cdot\|_2$

Here for $x\in\mathbb{R}^n$ we have:

$\|x\|_2=\sqrt{\sum^n_{i=1}x_i^2}$

TODO: proof

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 # <math>\forall x,y\in V\ \|x+y\|\le\|x\|+\|y\|</math> - a form of the [[Triangle inequality|triangle inequality]]
+Often parts 1 and 2 are combined into the statement
+* <math>\|x\|\ge 0\text{ and }\|x\|=0\iff x=0</math> so only 3 requirements will be stated.
+I don't like this
+==Examples==
+===The Euclidean Norm===
+The Euclidean norm is denoted <math>\|\cdot\|_2</math>
+Here for <math>x\in\mathbb{R}^n</math> we have:
+<math>\|x\|_2=\sqrt{\sum^n_{i=1}x_i^2}</math>
+{{Todo|proof}}
 {{Definition|Linear Algebra}}