Normed Space Smoothness

The most abundant part of the geometric theory of normed space is the geomet-ric theory of Banach spaces. The concept of smoothness in Banach spaces plays avery important role. Smoothness, on the one hand, is put forward by the convexityof the dual nature; On the other hand, it is closely linked to diferentiability of thenorm. In recent years, mathematicians have studied the smoothness and its point-wise nature, and obtained a lot of results. But for the pointwise properties of theextreme smoothness have not been yet studied carefully. These properties have rela-tions with Fre′chet diferentiability and Gateaux diferentiability in X.Meanwhile,alot of results characterizing smoothness, convexity by generalized orthogonality comeout.We discuss the existence and some nature of generalized orthogonality,and giveout the defnition and criterion of extreme smooth point in this paper.In chapter one, we recall the development of the Orlicz spaces and some result-s about the smoothness of Orlicz spaces, overview the development of generalizedorthogonality. And we Introduce the relationship between the smoothness and con-vexity, as well as generalized orthogonality. In the two-dimensional Orlicz sequencespace with the Orlicz norm and the Luxemburg norm, we give the examples express-ing the existence of Birkhof orthogonality and isosceles orthogonality.In chapter two, we focus on some basic properties of Birkhof orthogonality andisosceles orthogonality, and some consequences about smoothness and convexity,as well as generalized orthogonality. We conclude some consequences which arenot given proof procedures in the literature with analysis methods and geometricmethods.In chapter three, we introduce the knowledge of Orlicz sequences spaces withthe Luxemburg norm. Extreme point, very extreme point and uniformly extremepoint are frst introduced in this paper, and the criteria of the extreme smooth pointin Orlicz sequences spaces with the Luxemburg norm is given.