Opial property
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 behavior of nonlinear semigroups. The property is named after the Polish mathematician Zdzisław Opial.
Definitions
Let be a Banach space. X is said to have the Opial property if, whenever n∈N is a sequence in X converging weakly to some x0 ∈ X and x ≠ x0, it follows thatAlternatively, 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 n∈N is a sequence in X converging weakly-∗ to some x0 ∈ X and x ≠ x0, it follows that
or, as above,
A Banach space X is said to have the uniform Opial property if, for every c > 0, there exists an r > 0 such that
for every x ∈ X with ||x|| ≥ c and every sequence n∈N in X converging weakly to 0 and with
Examples
- Opial's theorem : Every Hilbert space has the Opial property.
- Sequence spaces,, have the Opial property.
- Van Dulst theorem : for every separable Banach space there is an equivalent norm that endows it with the Opial property.
- For uniformly convex Banach spaces, Opial property holds if and only if Delta-convergence coincides with weak convergence.