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.