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Since the p-adic integers are homeomorphic to the Cantor set, they form a compact set. Any global field K is a discrete additive subgroup of its adele ring, and the quotient space is compact. This was used in John Tate's thesis to allow harmonic analysis to be used in number theory.
(In the context of real analysis, these notions are equivalent: a set in Euclidean space is compact if and only if it is closed and bounded.) Briefly, a closed set contains all of its boundary points , while a set is bounded if there exists a real number such that the distance between any two points of the set is less than that number.
The theorem in real analysis draws the same conclusion for closed and bounded subsets of the set of real numbers. It states that a decreasing nested sequence ( C k ) k ≥ 0 {\displaystyle (C_{k})_{k\geq 0}} of non-empty, closed and bounded subsets of R {\displaystyle \mathbb {R} } has a non-empty intersection.
The set where they hold is called the Lebesgue set of , and points in the Lebesgue set are called Lebesgue points. Lebesgue integral Lebesgue integral. Lebesgue measure Lebesgue measure. Lelong Lelong number. Levi Levi's problem asks to show a pseudoconvex set is a domain of holomorphy. line integral Line integral. Liouville
Lemma: A closed subset of a compact set is compact. Let K be a closed subset of a compact set T in R n and let C K be an open cover of K. Then U = R n \ K is an open set and = {} is an open cover of T. Since T is compact, then C T has a finite subcover ′, that also covers the smaller set K.
A random compact set is а measurable function from а probability space (,,) into (, ()). Put another way, a random compact set is a measurable function K : Ω → 2 M {\displaystyle K\colon \Omega \to 2^{M}} such that K ( ω ) {\displaystyle K(\omega )} is almost surely compact and
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.
If is the real line, or -dimensional Euclidean space, then a function has compact support if and only if it has bounded support, since a subset of is compact if and only if it is closed and bounded. For example, the function f : R → R {\displaystyle f:\mathbb {R} \to \mathbb {R} } defined above is a continuous function with compact support ...