Opial property

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 x₀ ∈ X and x ≠ x₀, it follows that

${\displaystyle \liminf _{n\to \infty }\|x_{n}-x_{0}\|<\liminf _{n\to \infty }\|x_{n}-x\|.}$

Alternatively, using the contrapositive, this condition may be written as

${\displaystyle \liminf _{n\to \infty }\|x_{n}-x\|\leq \liminf _{n\to \infty }\|x_{n}-x_{0}\|\implies x=x_{0}.}$

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 x₀ ∈ X and x ≠ x₀, it follows that

${\displaystyle \liminf _{n\to \infty }\|x_{n}-x_{0}\|<\liminf _{n\to \infty }\|x_{n}-x\|,}$

or, as above,

${\displaystyle \liminf _{n\to \infty }\|x_{n}-x\|\leq \liminf _{n\to \infty }\|x_{n}-x_{0}\|\implies x=x_{0}.}$

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

${\displaystyle 1+r\leq \liminf _{n\to \infty }\|x_{n}-x\|}$

for every x ∈ X with ||x|| ≥ c and every sequence (x_n)_n∈N in X converging weakly (weakly-∗) to 0 and with

${\displaystyle \liminf _{n\to \infty }\|x_{n}\|\geq 1.}$

• Opial's theorem (1967): Every Hilbert space has the Opial property.
• Sequence spaces ${\displaystyle \ell ^{p}}$, ${\displaystyle 1\leq p<\infty }$, have the Opial property.
• Van Dulst theorem (1982): 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.

• Opial, Zdzisław (1967). "Weak convergence of the sequence of successive approximations for nonexpansive mappings". Bull. Amer. Math. Soc. 73 (4): 591–597. doi:10.1090/S0002-9904-1967-11761-0.