Definitions
Let (X, || ||) be a Banach space. X is said to have the Opial property if, whenever (xn)n∈N is a sequence in X converging weakly to some x0 ∈ X and x ≠ x0, 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 (xn)n∈N is a sequence in X converging weakly-∗ to some x0 ∈ X and x ≠ x0, 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 (xn)n∈N in X converging weakly (weakly-∗) to 0 and with
Read more about this topic: Opial Property
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