Locally uniform Opial conditions Original Research Article

A Banach space X is said to have the locally uniform Opial condition if for every weakly null sequence {xn} in X with View the MathML source and any c>0 there is r>0 such that View the MathML source for every x∈X with ||x||⩾c. Lin, Tan and Xu assert that it is possible to replace “View the MathML source” with “View the MathML source” in this definition without affecting the meaning. It is shown here that the two concepts are in general different but that in Banach spaces having a separable dual, the condition with “View the MathML source” is equivalent to the uniform Opial property.