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| '''Unconditional convergence''' is a topological property (convergence) related to an algebraical object (sum). It is an extension of the notion of convergence for series of countably many elements to series of arbitrarily many. It has been mostly studied in [[Banach space]]s. | | The writer is called Irwin Wunder but it's not the most masucline name out there. Hiring has been my occupation for some time but I've currently applied for another one. Years ago we moved to Puerto Rico and my family members loves it. Doing ceramics is what adore performing.<br><br>Review my homepage; [http://www.1a-pornotube.com/blog/84958 std testing at home] |
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| == Definition ==
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| Let <math>X</math> be a [[topological vector space]]. Let <math>I</math> be an [[index set]] and <math>x_i \in X</math> for all <math>i \in I</math>.
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| The series <math>\textstyle \sum_{i \in I} x_i</math> is called '''unconditionally convergent''' to <math>x \in X</math>, if
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| * the indexing set <math>I_0 :=\{i\in I: x_i\ne 0\}</math> is [[countable]] and
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| * for every [[permutation]] of <math>I_0 :=\{i\in I: x_i\ne 0\}</math> the relation holds:<math>\sum_{i=1}^\infty x_i = x</math>
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| ==Alternative definition==
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| Unconditional convergence is often defined in an equivalent way: A series is unconditionally convergent if for every sequence <math>(\varepsilon_n)_{n=1}^\infty</math>, with <math>\varepsilon_n\in\{-1, +1\}</math>, the series
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| :<math>\sum_{n=1}^\infty \varepsilon_n x_n</math>
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| converges.
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| Every [[absolute convergence|absolutely convergent]] series is unconditionally convergent, but the [[converse (logic)|converse]] implication does not hold in general. When ''X'' = '''R'''<sup>''n''</sup>, then, by the [[Riemann series theorem]], the series <math>\sum x_n</math> is unconditionally convergent if and only if it is absolutely convergent.
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| ==See also==
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| *[[Modes of convergence (annotated index)]]
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| ==References==
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| * Ch. Heil: [http://www.math.gatech.edu/~heil/papers/bases.pdf A Basis Theory Primer]
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| * K. Knopp: "Theory and application of infinite series"
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| * K. Knopp: "Infinite sequences and series"
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| * P. Wojtaszczyk: "Banach spaces for analysts"
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| {{PlanetMath attribution|id=7358|title=Unconditional convergence}}
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| [[Category:Mathematical analysis]]
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| [[Category:Mathematical series]]
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| [[Category:Summability theory]]
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| [[Category:Convergence (mathematics)]]
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The writer is called Irwin Wunder but it's not the most masucline name out there. Hiring has been my occupation for some time but I've currently applied for another one. Years ago we moved to Puerto Rico and my family members loves it. Doing ceramics is what adore performing.
Review my homepage; std testing at home