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| In [[mathematics]], an element ''z'' of a [[Banach algebra]] ''A'' is called a '''topological divisor of zero''' if there exists a [[sequence]] ''x''<sub>1</sub>, ''x''<sub>2</sub>, ''x''<sub>3</sub>, ... of elements of ''A'' such that
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| # The sequence ''zx''<sub>''n''</sub> converges to the zero element, but
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| # The sequence ''x''<sub>''n''</sub> does not converge to the zero element.
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| If such a sequence exists, then one may assume that ||''x''<sub>''n''</sub>|| = 1 for all ''n''.
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| If ''A'' is not [[commutativity|commutative]], then ''z'' is called a '''left''' topological divisor of zero, and one may define right topological divisors of zero similarly.
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| ==Examples==
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| * If ''A'' has a unit element, then the invertible elements of ''A'' form an [[open set|open subset]] of ''A'', while the non-invertible elements are the complementary [[closed set|closed subset]]. Any point on the [[boundary (topology)|boundary]] between these two sets is both a left and right topological divisor of zero.
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| * In particular, any [[quasinilpotent]] element is a topological divisor of zero (e.g. the [[Volterra operator]]).
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| * An operator on a Banach space <math>X</math>, which is [[injective]], not [[surjective]], but whose image is dense in <math>X</math>, is a left topological divisor of zero.
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| ==Generalization==
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| The notion of a topological divisor of zero may be generalized to any [[topological algebra]]. If the algebra in question is not [[first-countable space|first-countable]], one must substitute [[net (mathematics)|nets]] for the sequences used in the definition.
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| {{Unreferenced|date=January 2011}}
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| {{DEFAULTSORT:Topological Divisor Of Zero}}
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| [[Category:Topological algebra]]
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| {{algebra-stub}}
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Nice to meet you, my title is Refugia. To gather cash is what her family members and her enjoy. Managing people is his occupation. Years in the past we moved to North Dakota.
Visit my homepage; home std test kit, Suggested Studying,