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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''1, ''x''2, ''x''3, ... of elements of ''A'' such that # The sequence ''zx''''n'' converges to the zero element, but # The sequence ''x''''n'' does not converge to the zero element. If such a sequence exists, then one may assume that ||''x''''n''|| = 1 for all ''n''. If ''A'' is not commutative, then ''z'' is called a left topological divisor of zero, and one may define right topological divisors of zero similarly. ==Examples== * If ''A'' has a unit element, then the invertible elements of ''A'' form an open subset of ''A'', while the non-invertible elements are the complementary closed subset. Any point on the boundary between these two sets is both a left and right topological divisor of zero. * In particular, any quasinilpotent element is a topological divisor of zero (e.g. the Volterra operator). * An operator on a Banach space , which is injective, not surjective, but whose image is dense in , is a left topological divisor of zero. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Topological divisor of zero」の詳細全文を読む スポンサード リンク
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