Cryptanalysis of the Lorenz cipher

Template:No footnotes In functional analysis, the Fréchet-Kolmogorov theorem (the names of Riesz or Weil are sometimes added as well) gives a necessary and sufficient condition for a set of functions to be relatively compact in an L^p space. It can be thought of as an L^p version of the Arzelà-Ascoli theorem, from which it can be deduced.

Statement

Let $B$ be a bounded set in $L^p({\mathbb{ℝ}}^n)$, with $p\in [1,\mathrm{\infty })$.

The subset B is relatively compact if and only if the following properties hold:

1. $\underset{r\to \mathrm{\infty }}{\lim }\underset{\left|x\right|>r}{\int }{\left|f\right|}^p=0$ uniformly on B,
2. $\underset{a\to 0}{\lim }\Vert {\tau }_{a}f-f{\Vert }_{L^p({\mathbb{ℝ}}^n)}=0$ uniformly on B,

where ${\tau }_{a}f$ denotes the translation of $f$ by $a$, that is, ${\tau }_{a}f(x)=f(x-a).$

The second property can be stated as $\mathrm{\forall }\epsilon >0\mathrm{\exists }\delta >0$ such that $\Vert {\tau }_{a}f-f{\Vert }_{L^p({\mathbb{ℝ}}^n)}<\epsilon \mathrm{\forall }f\in B,\mathrm{\forall }a$ with $\left|a\right|<\delta .$

References

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• Marcel Riesz, « Sur les ensembles compacts de fonctions sommables », dans Acta Sci. Math., vol. 6, 1933, p. 136–142
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
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