A pythagorean approach in Banach spaces. (3rd January 2006) - Gao, Ji
Abstract : LetX be a Banach space and letS ( X ) = x ∈ X, ‖ x ‖ = 1 be the unit sphere ofX . ParametersE ( X ) = sup ⁡ α ( x ), x ∈ S ( X ) , e ( X ) = inf ⁡ α ( x ), x ∈ S ( X ) , F ( X ) = sup ⁡ β ( x ), x ∈ S ( X ) , andf ( X ) = inf ⁡ β ( x ), x ∈ S ( X ) , whereα ( x ) = sup ⁡ ‖ x + y ‖ 2 + ‖ x − y ‖ 2, y ∈ S ( X ) andβ ( x ) = inf ⁡ ‖ x + y ‖ 2 + ‖ x − y ‖ 2, y ∈ S ( X ) are introduced and studied. The values of these parameters in thel p spaces and function spacesL p [ 0, 1 ] are estimated. Among the other results, we proved that a Banach spaceX withE ( X ) < 8, orf ( X ) > 2 is uniform nonsquare; and a Banach spaceX withE ( X ) < 5, orf ( X ) > 32 / 9 has uniform normal structure.