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| In [[mathematics]], '''Helly's selection theorem''' states that a [[sequence]] of functions that is locally of [[Bounded variation|bounded total variation]] and [[uniformly bounded]] at a point has a [[Convergent sequence|convergent]] [[subsequence]]. In other words, it is a [[compactness theorem]] for the space BV<sub>loc</sub>. It is named for the [[Austria]]n [[mathematician]] [[Eduard Helly]].
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| The theorem has applications throughout [[mathematical analysis]]. In [[probability theory]], the result implies compactness of a [[tightness of measures|tight family of measures]].
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| ==Statement of the theorem==
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| Let ''U'' be an [[open subset]] of the [[real line]] and let ''f''<sub>''n''</sub> : ''U'' → '''R''', ''n'' ∈ '''N''', be a sequence of functions. Suppose that
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| * (''f''<sub>''n''</sub>) has uniformly [[Bounded variation|bounded]] [[total variation]] on any ''W'' that is [[compactly embedded]] in ''U''. That is, for all sets ''W'' ⊆ ''U'' with [[compact space|compact]] [[closure (topology)|closure]] ''W̄'' ⊆ ''U'',
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| ::<math>\sup_{n \in \mathbb{N}} \left( \left\| f_{n} \right\|_{L^{1} (W)} + \left\| \frac{\mathrm{d} f_{n}}{\mathrm{d} t} \right\|_{L^{1} (W)} \right) < + \infty,</math>
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| :where the derivative is taken in the sense of [[Distribution (mathematics)|tempered distributions]];
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| * and (''f''<sub>''n''</sub>) is uniformly bounded at a point. That is, for some ''t'' ∈ ''U'', { ''f''<sub>''n''</sub>(''t'') | ''n'' ∈ '''N''' } ⊆ '''R''' is a [[bounded set]].
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| Then there exists a [[subsequence]] ''f''<sub>''n''<sub>''k''</sub></sub>, ''k'' ∈ '''N''', of ''f''<sub>''n''</sub> and a function ''f'' : ''U'' → '''R''', locally of [[bounded variation]], such that
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| * ''f''<sub>''n''<sub>''k''</sub></sub> converges to ''f'' pointwise;
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| * and ''f''<sub>''n''<sub>''k''</sub></sub> converges to ''f'' locally in ''L''<sup>1</sup> (see [[locally integrable function]]), i.e., for all ''W'' compactly embedded in ''U'',
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| ::<math>\lim_{k \to \infty} \int_{W} \big| f_{n_{k}} (x) - f(x) \big| \, \mathrm{d} x = 0;</math>
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| * and, for ''W'' compactly embedded in ''U'',
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| ::<math>\left\| \frac{\mathrm{d} f}{\mathrm{d} t} \right\|_{L^{1} (W)} \leq \liminf_{k \to \infty} \left\| \frac{\mathrm{d} f_{n_{k}}}{\mathrm{d} t} \right\|_{L^{1} (W)}. </math>
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| ==Generalizations==
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| There are many generalizations and refinements of Helly's theorem. The following theorem, for BV functions taking values in [[Banach space]]s, is due to Barbu and Precupanu:
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| Let ''X'' be a [[reflexive space|reflexive]], [[separable space|separable]] Hilbert space and let ''E'' be a closed, [[convex set|convex]] subset of ''X''. Let Δ : ''X'' → [0, +∞) be [[positive-definite]] and [[homogeneous function|homogeneous of degree one]]. Suppose that ''z''<sub>''n''</sub> is a uniformly bounded sequence in BV([0, ''T'']; ''X'') with ''z''<sub>''n''</sub>(''t'') ∈ ''E'' for all ''n'' ∈ '''N''' and ''t'' ∈ [0, ''T'']. Then there exists a subsequence ''z''<sub>''n''<sub>''k''</sub></sub> and functions ''δ'', ''z'' ∈ BV([0, ''T'']; ''X'') such that
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| * for all ''t'' ∈ [0, ''T''],
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| ::<math>\int_{[0, t)} \Delta (\mathrm{d} z_{n_{k}}) \to \delta(t);</math>
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| * and, for all ''t'' ∈ [0, ''T''],
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| ::<math>z_{n_{k}} (t) \rightharpoonup z(t) \in E;</math>
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| * and, for all 0 ≤ ''s'' < ''t'' ≤ ''T'',
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| ::<math>\int_{[s, t)} \Delta(\mathrm{d} z) \leq \delta(t) - \delta(s).</math>
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| == See also ==
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| * [[Bounded variation]]
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| * [[Fraňková-Helly selection theorem]]
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| * [[Total variation]]
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| ==References==
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| * {{cite book
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| | last = Barbu
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| | first = V.
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| | coauthors = Precupanu, Th.
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| | title = Convexity and optimization in Banach spaces
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| | series = Mathematics and its Applications (East European Series)
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| | volume = 10
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| | edition = Second Romanian Edition
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| | publisher = D. Reidel Publishing Co.
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| | location = Dordrecht
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| | year = 1986
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| | isbn = 90-277-1761-3
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| | nopp = true
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| | page = xviii+397
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| }} {{MathSciNet|id=860772}}
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| [[Category:Compactness theorems]]
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| [[Category:Theorems in analysis]]
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She is known by the title of Myrtle Shryock. My family members lives in Minnesota and my family enjoys it. What I love doing is to gather badges but I've been taking on new issues lately. Bookkeeping is my occupation.
Take a look at my page: std home test