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In mathematics, the Opial property is an abstract property of Banach spaces that plays an important role in the study of weak convergence of iterates of mappings of Banach spaces, and of the asymptotic behaviour of nonlinear semigroups. The property is named after the Polish mathematician Zdzisław Opial. ==Definitions== Let (''X'', || ||) be a Banach space. ''X'' is said to have the Opial property if, whenever (''x''''n'')''n''∈N is a sequence in ''X'' converging weakly to some ''x''0 ∈ ''X'' and ''x'' ≠ ''x''0, it follows that : Alternatively, using the contrapositive, this condition may be written as : If ''X'' is the continuous dual space of some other Banach space ''Y'', then ''X'' is said to have the weak-∗ Opial property if, whenever (''x''''n'')''n''∈N is a sequence in ''X'' converging weakly-∗ to some ''x''0 ∈ ''X'' and ''x'' ≠ ''x''0, it follows that : or, as above, : A (dual) Banach space ''X'' is said to have the uniform (weak-∗) Opial property if, for every ''c'' > 0, there exists an ''r'' > 0 such that : for every ''x'' ∈ ''X'' with ||''x''|| ≥ c and every sequence (''x''''n'')''n''∈N in ''X'' converging weakly (weakly-∗) to 0 and with : 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「Opial property」の詳細全文を読む スポンサード リンク
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