Théorème de sélection de Helly

Helly's selection theorem In mathematics, Théorème de sélection de Helly (also called the Helly selection principle) states that a uniformly bounded sequence of monotone real functions admits a convergent subsequence. Autrement dit, it is a sequential compactness theorem for the space of uniformly bounded monotone functions. It is named for the Austrian mathematician Eduard Helly. A more general version of the theorem asserts compactness of the space BVloc of functions locally of bounded total variation that are uniformly bounded at a point.

The theorem has applications throughout mathematical analysis. En théorie des probabilités, the result implies compactness of a tight family of measures.

Contenu 1 Énoncé du théorème 2 Generalisation to BVloc 3 Autres généralisations 4 Voir également 5 References Statement of the theorem Let (fn)n ∈ N be a sequence of increasing functions mapping the real line R into itself, and suppose that it is uniformly bounded: there are a,b ∈ R such that a ≤ fn ≤ b for every n ∈ N. Then the sequence (fn)n ∈ N admits a pointwise convergent subsequence.

Generalisation to BVloc Let U be an open subset of the real line and let fn : U → R, n ∈ N, be a sequence of functions. Supposer que (fn) has uniformly bounded total variation on any W that is compactly embedded in U. C'est-à-dire, for all sets W ⊆ U with compact closure W̄ ⊆ U, {displaystyle sup _{nin mathbf {N} }la gauche(la gauche|F_{n}droit|_{L^{1}(O)}+la gauche|{frac {mathrm {ré} F_{n}}{mathrm {ré} t}}droit|_{L^{1}(O)}droit)<+infty ,} where the derivative is taken in the sense of tempered distributions; and (fn) is uniformly bounded at a point. That is, for some t ∈ U, { fn(t) | n ∈ N } ⊆ R is a bounded set. Then there exists a subsequence fnk, k ∈ N, of fn and a function f : U → R, locally of bounded variation, such that fnk converges to f pointwise; and fnk converges to f locally in L1 (see locally integrable function), i.e., for all W compactly embedded in U, {displaystyle lim _{kto infty }int _{W}{big |}f_{n_{k}}(x)-f(x){big |},mathrm {d} x=0;} and, for W compactly embedded in U, {displaystyle left|{frac {mathrm {d} f}{mathrm {d} t}}right|_{L^{1}(W)}leq liminf _{kto infty }left|{frac {mathrm {d} f_{n_{k}}}{mathrm {d} t}}right|_{L^{1}(W)}.} Further generalizations There are many generalizations and refinements of Helly's theorem. The following theorem, for BV functions taking values in Banach spaces, is due to Barbu and Precupanu: Let X be a reflexive, separable Hilbert space and let E be a closed, convex subset of X. Let Δ : X → [0, +∞) be positive-definite and homogeneous of degree one. Suppose that zn is a uniformly bounded sequence in BV([0, T]; X) with zn(t) ∈ E for all n ∈ N and t ∈ [0, T]. Then there exists a subsequence znk and functions δ, z ∈ BV([0, T]; X) such that for all t ∈ [0, T], {displaystyle int _{[0,t)}Delta (mathrm {d} z_{n_{k}})to delta (t);} and, for all t ∈ [0, T], {displaystyle z_{n_{k}}(t)rightharpoonup z(t)in E;} and, for all 0 ≤ s < t ≤ T, {displaystyle int _{[s,t)}Delta (mathrm {d} z)leq delta (t)-delta (s).} See also Bounded variation Fraňková-Helly selection theorem Total variation References Rudin, W. (1976). Principles of Mathematical Analysis. International Series in Pure and Applied Mathematics (Third ed.). New York: McGraw-Hill. 167. ISBN 978-0070542358. Barbu, V.; Precupanu, Th. (1986). Convexity and optimization in Banach spaces. Mathematics and its Applications (East European Series). Vol. 10 (Second Romanian ed.). Dordrecht: D. Reidel Publishing Co. xviii+397. ISBN 90-277-1761-3. MR860772 Categories: Compactness theoremsTheorems in analysis

Si vous voulez connaître d'autres articles similaires à Théorème de sélection de Helly vous pouvez visiter la catégorie Théorèmes de compacité.

Laisser un commentaire

Votre adresse email ne sera pas publiée.


Nous utilisons nos propres cookies et ceux de tiers pour améliorer l'expérience utilisateur Plus d'informations