Difference between revisions of "Norm"
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(Created page with "==Definition== A norm on a vector space {{M|(V,F)}} is a function <math>\|\cdot\|:V\rightarrow\mathbb{R}</math> such that: # <math>\forall x\in V\ \|x\|\ge 0<...") |
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# <math>\|x\|=0\iff x=0</math> | # <math>\|x\|=0\iff x=0</math> | ||
# <math>\forall \lambda\in F, x\in V\ \|\lambda x\|=|\lambda|\|x\|</math> where <math>|\cdot|</math> denotes [[Absolute value|absolute value]] | # <math>\forall \lambda\in F, x\in V\ \|\lambda x\|=|\lambda|\|x\|</math> where <math>|\cdot|</math> denotes [[Absolute value|absolute value]] | ||
| + | # <math>\forall x,y\in V\ \|x+y\|\le\|x\|+\|y\|</math> - a form of the [[Triangle inequality|triangle inequality]] | ||
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{{Definition|Linear Algebra}} | {{Definition|Linear Algebra}} | ||
Revision as of 16:13, 7 March 2015
Definition
A norm on a vector space (V,F) is a function ∥⋅∥:V→R such that:
- ∀x∈V ∥x∥≥0
- ∥x∥=0⟺x=0
- ∀λ∈F,x∈V ∥λx∥=|λ|∥x∥ where |⋅| denotes absolute value
- ∀x,y∈V ∥x+y∥≤∥x∥+∥y∥ - a form of the triangle inequality
