A NOTE ON NORM ATTAINING FUNCTIONALS

We are concerned in this paper with the density of functionals which d o not attain their norms in Banach spaces. Some earlier results given for separable spaces are extended to the nonseparable case. We obtain that a Banach space X is reflexive if and only if it satisfies any of the following properties: (i) X admits a norm \\.\\ with the Mazur Int ersection Property and the set NA(\\.\\) of all norm attaining functio nals of X contains an open set, (ii) the set NA(\\.\\)(1) of all norm one elements of NA(\\.\\) contains a (relative) weak open set of the unit sphere, (iii) X has C*PCP and NA(\\.\\)(1) contains a (relative ) weak open set of the unit sphere, (iv) X is WCG, X has CPCP and NA( \\.\\)(1) contains a (relative) weak open set of the unit sphere. Fina lly, if X is separable, then X is reflexive if and only if NA(\\.\\)(1 ) Contains a (relative) weak open set of the unit sphere.