P-Norm
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A P-Norm is a generalization of the mathematical norm for a n-dimensional vector space which statisfies the triangle inequality.
- Context:
- It is defined as [math]\displaystyle{ \left\| \mathbf{x} \right\| _p = \bigg( \sum_{i=1}^n \left| x_i \right| ^p \bigg) ^{1/p} }[/math], where [math]\displaystyle{ p }[/math] is greater or equal to 1 and real.
- It can be used to defined the norm a [math]\displaystyle{ m \times n }[/math] matrix ([math]\displaystyle{ A }[/math]) as [math]\displaystyle{ \Vert A \Vert_p =\left( \sum_{i=1}^m \sum_{j=1}^n |a_{ij}|^p \right)^{1/p} }[/math]
- Example(s):
- Counter-Example(s):
- See: Lp Space, Vector Norm, Mathematical Norm, Euclidean Norm.
References
2015
- (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/Norm_(mathematics)#p-norm
- QUOTE: Let p ≥ 1 be a real number.
- [math]\displaystyle{ \left\| \mathbf{x} \right\| _p := \bigg( \sum_{i=1}^n \left| x_i \right| ^p \bigg) ^{1/p}. }[/math]
- For p = 1 we get the taxicab norm, for p = 2 we get the Euclidean norm, and as p approaches [math]\displaystyle{ \infty }[/math] the p-norm approaches the infinity norm or maximum norm. The p-norm is related to the generalized mean or power mean.
- This definition is still of some interest for 0 < p < 1, but the resulting function does not define a norm, because it violates the triangle inequality. What is true for this case of 0 < p < 1, even in the measurable analog, is that the corresponding Lp class is a vector space, and it is also true that the function
- [math]\displaystyle{ \int_X \left| f (x ) - g (x ) \right| ^p ~ \mathrm d \mu }[/math]
- (without pth root) defines a distance that makes Lp(X) into a complete metric topological vector space. These spaces are of great interest in functional analysis, probability theory, and harmonic analysis.
- However, outside trivial cases, this topological vector space is not locally convex and has no continuous nonzero linear forms. Thus the topological dual space contains only the zero functional.