P-Norm

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):
  • Euclidean Norm.
  • Trace Norm.
  • Schatten Norm.
  • …
• Counter-Example(s):
  • Zero-Norm.
• See: Lp Space, Vector Norm, Mathematical Norm, Euclidean Norm.

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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 L^p 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 L^p(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.

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