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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 topology|weak convergence]] of iterates of mappings of Banach spaces, and of the asymptotic behaviour of nonlinear [[semigroup]]s. The property is named after the [[Poland|Polish]] [[mathematician]] [[Zdzisław Opial]].
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| ==Definitions==
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| Let (''X'', || ||) be a Banach space. ''X'' is said to have the '''Opial property''' if, whenever (''x''<sub>''n''</sub>)<sub>''n''∈'''N'''</sub> is a sequence in ''X'' converging weakly to some ''x''<sub>0</sub> ∈ ''X'' and ''x'' ≠ ''x''<sub>0</sub>, it follows that
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| :<math>\liminf_{n \to \infty} \| x_{n} - x_{0} \| < \liminf_{n \to \infty} \| x_{n} - x \|.</math>
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| Alternatively, using the [[contrapositive]], this condition may be written as
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| :<math>\liminf_{n \to \infty} \| x_{n} - x \| \leq \liminf_{n \to \infty} \| x_{n} - x_{0} \| \implies x = x_{0}.</math>
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| 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''<sub>''n''</sub>)<sub>''n''∈'''N'''</sub> is a sequence in ''X'' converging weakly-∗ to some ''x''<sub>0</sub> ∈ ''X'' and ''x'' ≠ ''x''<sub>0</sub>, it follows that
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| :<math>\liminf_{n \to \infty} \| x_{n} - x_{0} \| < \liminf_{n \to \infty} \| x_{n} - x \|,</math> | |
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| or, as above,
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| :<math>\liminf_{n \to \infty} \| x_{n} - x \| \leq \liminf_{n \to \infty} \| x_{n} - x_{0} \| \implies x = x_{0}.</math> | |
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| 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
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| :<math>1 + r \leq \liminf_{n \to \infty} \| x_{n} - x \|</math> | |
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| for every ''x'' ∈ ''X'' with ||''x''|| ≥ c and every sequence (''x''<sub>''n''</sub>)<sub>''n''∈'''N'''</sub> in ''X'' converging weakly (weakly-∗) to 0 and with
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| :<math>\liminf_{n \to \infty} \| x_{n} \| \geq 1.</math>
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| ==Examples==
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| * Opial's theorem (1967): Every [[Hilbert space]] has the Opial property.
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| ==References==
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| * {{cite journal
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| | last = Opial
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| | first = Zdzisław
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| | title = Weak convergence of the sequence of successive approximations for nonexpansive mappings
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| | journal = Bull. Amer. Math. Soc.
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| | volume = 73
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| | year = 1967
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| | pages = 591–597
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| | doi = 10.1090/S0002-9904-1967-11761-0
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| | issue = 4
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| }}
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| [[Category:Banach spaces]]
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