L-Infinity Space

A L-Infinity Space is a function space or a sequence space which elements are bounded sequences or bounded functions, respectively.

• AKA: L-Infinity.
• Context:
  • It is usually denoted as [math]\displaystyle{ L_\infty }[/math] in the context of a function space and as [math]\displaystyle{ \ell^{\infty} }[/math] in context of a sequence space.
• Example(s):
  • Lp Space when [math]\displaystyle{ p\rightarrow \infty }[/math]
• Counter-Example(s):
  • Euclidean Space.
  • Banach Space.
• See: L1-Norm, L2-Norm, Lp Space, L-Infinity Norm, Sequence Space, Function Space, Banach Space.

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References

2015

• (Wikipedia, 2015) ⇒
  • QUOTE: : Sequence Space, ℓ^∞ is a sequence space. Its elements are the bounded sequences. The vector space operations, addition and scalar multiplication, are applied coordinate by coordinate. ℓ^∞ is the largest ℓp space. With respect to the norm [math]\displaystyle{ \|x\|_\infty = \sum_n |x_n|, }[/math] ℓ^∞ is also a Banach space.

Function Space, [math]\displaystyle{ L_\infty }[/math] is a function space. Its elements are the essentially bounded measurable functions. More precisely, [math]\displaystyle{ L_\infty }[/math] is defined based on an underlying measure space, (S, Σ, μ). Start with the set of all measurable functions from S to R which are essentially bounded, i.e. bounded up to a set of measure zero. Two such functions are identified if they are equal almost everywhere. Denote the resulting set by L^∞(S, μ).

For a function  f  in this set, its essential supremum serves as an appropriate norm:

[math]\displaystyle{ \|f\|_\infty \equiv \inf \{ C\ge 0 : |f(x)| \le C \text{ for almost every } x \}. }[/math]

2011

• (Wilkinson et al., 2011) ⇒ Leland Wilkinson, Anushka Anand, and Dang Nhon Tuan. (2011). “CHIRP: A New Classifier based on Composite Hypercubes on Iterated Random Projections.” In: Proceedings of the 17th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (KDD-2011) Journal. ISBN:978-1-4503-0813-7 doi:10.1145/2020408.2020418

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