The Smooth Bishop-Phelps-Bollobás Theorem

The celebrated Bishop-Phelps theorem([34])states that the set of norm attaining func-tionals on a Banach space is norm dense in the dual space.The variational principles(such as Ekeland variational principle([16,18])etc.)and its applications based on this theorem have become the cornerstone of many branches of modern Mathematics and Applied Mathe-matrics.For more than 50 years,various of its quantitative presentation have become a new research field of mathematrics.However,the more precise presentation under the situation of"smoothness" has not aroused the academic attention it deserves yet.For this reason,we introduce the concept of“relatively smooth point”in this paper,and then prove a smoothly quantitative version of the Bishop-Phelps-Bollobás theorem al-ways holds in a class of Banach spaces including all separable Banach spaces:Supposed that X is a Banach space,and there is a strictly convex dual norm on X*.Then for any ε>0,x0 ∈ SX and x0*∈ SX*satisfying 〈x0*,x0〉＞ 1—ε2/4,there exist xε ∈ SX and Xε*∈ SX*such that〈xε*,xε〉 = 1,||xε－x0|| ≤ ε and ||xε*－x0*||≤ε.where xε is a ball-relative smooth point,and xε*is the ball-relative Gateaux derivative of the norm at x,.We also prove the following strongly smooth Bishop-Phelps-Bollobás theorem:Supposed that X is a separable Banach space,and its dual space X*is also a separable Banach space.Then for any ε>0,x0 ∈ SX and xo*∈ SX*satisfying 〈x0*,X0〉＞ 1—ε 2/4,then there existXε*∈ SX and Xε*∈SX*such that〈xε*,xε〉 = 1,||xε－x0|| ≤ ε and||xε*－x0*||≤ε.where xε is a ball-relative strongly smooth point,and xε*is the ball-relative Frechet deriva-tive of the norm at xε.