**S.S. Dragomir
**

*Abstract*

Semi-Inner Products, that can be naturally defined in general Banach spaces over the
real or complex number field, play an important role in describing the geometric
properties of these spaces. In the first chapter of the book, a short introduction to the main properties of the duality mapping that will be used in the next chapters is given. Chapter 2 is devoted to the semi-inner products in the sense of Lumer-Giles while the 3rd chapter is concerning with the main properties of the superior and inferior semi-inner products. In the next chapter the main properties of Milicic’s semi-inner product and the properties of normed spaces of type are presented. The next two chapters investigate the geometric properties of
(G)-, (Q) and (SQ)-inner product spaces introduced by the author, while Chapter 7 is
entirely devoted to the study of different mappings that can naturally be associated to
the norm derivatives in general normed spaces and, in particular, in inner product
spaces. Chapters 8 and 9 investigate different orthogonalities that may be introduced in
normed spaces and their intimate relationship with semi-inner products. In Chapter 11,
orthogonal decomposition theorems in general normed spaces are provided, while in the
next chapter the problem of approximating continuous linear functionals in general
normed spaces and characterizations of reflexivity in this context are given. A deeper
insight on this problem is then considered in Chapter 13, where some classes of
continuous functionals are introduced and a density result based on the famous
Bishop-Phelps theorem is obtained. In Chapter 14, the class of smooth normed spaces
of 2k-type and their application for non-linear operators is presented. In the next
chapter the continuous sublinear functionals defined in Reflexive Banach spaces is
investigated, while Chapter 16 deals with convex functions defined in more general
spaces endowed with subinner products. The monograph concludes by considering the
representation problem of linear forms defined on modules endowed with general
semi-subinner products.
(BD)
Alpha version: 20th January, 2003. |