The present book is concerned with various semi-inner products on normed spaces, with applications to the study of the geometrical properties of these spaces, and the representation of continuous linear functionals in terms of semi-inner products (analogs of Riesz’s representation theorem in Hilbert spaces).
The notion of a semi-inner product (s.i.p. for short) was defined by G. Lumer in 1961 and extensively studied in subsequent papers by J. R. Giles, P.-L. Papini, P. M. Miličići, I. Roşca, and others. For a real or complex normed linear space \((X,\| \,\| )\), a Lumer-Giles s.i.p. (LG-s.i.p.) is a mapping \([\,,\,]:X^ 2\to \mathbb K,\; \mathbb K =\mathbb R \) or \(\mathbb C, \) such that \([x,x] > 0\) for \(x\neq 0,\) which is linear in the first variable, \(\,[x,\lambda y] = \bar \lambda [x,y], \,\) and satisfies Schwarz’s inequality \(\, | [x,y]| ^ 2 \leq [x,x][y,y],\,\) for all \(x,y\in X\) and \(\lambda \in \mathbb K.\) The mapping \(\| x\| _ 1 = [x,x]^{1/2},\; x\in X,\,\) is a norm on \(X\) and \(f_y(x) = [x,y],\;x\in X,\,\) is a continuous linear functional on \((X,\| \,\| _ 1)\) of norm \(\| f_y\| =\| y\| _ 1.\) As it was shown by I. Roşca (1976), any LG-s.i.p. on a normed space \((X,\| \,\| )\) can be represented by means of the duality mapping \(J(x) =\{x^ * \in X^ * : \langle x^ *,x\rangle = \| x^ *\| \| x\| \;\) and \(\; \| x^ *\| = \| x\| \} \) in the form \([x,y] =\langle \widetilde J(x),y\rangle,\; x,y\in X,\,\) where \(\widetilde J\) is an arbitrary selection of the duality mapping \(J\).
The properties of the multivalued duality mapping \(J\) are briefly examined in the first chapter of the book. The LG-s.i.p. is studied in the second chapter, including characterizations of strict convexity and smoothness in terms of LG-s.i.p. The superior and the inferior s.i.p. defined by the formulae \(\, (x,y)_ i =\lim_{t\to 0\pm}\left(\| x+ty\| ^ 2-\| x\| ^ 2\right)/t,\,\) respectively, were studied by P. M. Miličić, R. A. Tapia, N. Pavel and I. Roşca. In the real case, we have \[ (x,y)_ i = \inf \{ [x,y] : [\,,\,] \in \mathcal S\} = \inf\{x^ *(y) : x^ *\in J(x)\}, \] where \(\mathcal S\) stands for the set of all s.i.p. on \(X\), and similar equalities for \((x,y)_ s\) but with \(\inf\) replaced by \(\sup\), justifying the terms superior and inferior included in their names. P. M. Milič ić also considered the mean value of these s.i.p., \(\, (x,y)_ g = \frac{1}{2}[(x,y)_ i+(x,y)_ s],\, \) called M-s.i.p. and studied in the fourth chapter. Chapter 5 is dedicated to a four-variable notion of s.i.p. due to the author, called Q-s.i.p., acting on \(X^ 4 \) and satisfying appropriate conditions inspired by those satisfied by the usual s.i.p. If the norm of a normed space \(X\) is generated by a Q-s.i.p., then \(X\) must be uniformly smooth. Further extensions of this notion are given in the next chapter (Chapter 5), where \(2k\)-inner products satisfying some generalized parallelogram law are defined and studied.
As in the case of usual inner-product spaces, an s.i.p. gives rise to a notion of orthogonality. The author studies in Chapters 8-12 the properties of orthogonalities associated with LG-s.i.p. and M-s.i.p., in connection with the classical orthogonality notions of Birkhoff and James.
The approximation of continuous linear functionals by linear functionals defined by s.i.p. is studied in Chapters 12 and 13. The cases of sublinear functionals and of convex functions are treated in Chapters 15 and 16, respectively. The last chapter of the book, Chapter 17, deals with representation results for linear functionals in terms of s.i.p.
Largely based on the author’s original results, the book collects a lot of useful properties of semi-inner products and of their generalizations, with emphasis on their relevance for the geometry of the underlying normed space.
The book will be useful to researchers in Banach space theory as well as for those working in linear or nonlinear operator equations and optimization. It can also be used as supplementary material for advanced courses in functional analysis.