Geometry of cones: in normed spaces (Q2832005)

The author is a well-known Soviet mathematician. This book was translated from Russian into German by Martin Weber.NEWLINENEWLINEThe main subject is the study of the geometry of a cone in normed spaces. Of course, this study is motivated by the existence of an ordered cone in many function spaces, for instance, the space of continuous functions \(C([0,1])\) on the unit interval of the real axis.NEWLINENEWLINEThe book has two parts: 1.~Introduction to the theoryNEWLINEof cones in normed spaces (Chapters I--V); 2.~Special problems of the geometry of cones in normed spaces (Chapters VI - XI).NEWLINENEWLINEThe first chapter -- Ordered vector spaces -- is dedicated to fundamental notions of ordered spaces like cones, partial order in vector spaces, order convergence, vector lattices, positive linear functionals, partial order in a dual space.NEWLINENEWLINESolid cones and closed cones are the subject of Chapter~2. It should be mentioned that the terminology is different from that used nowadays: for instance, solid cone means a cone with an inner point. The study of these cones is made in great detail, a possible explanation of this being that this book was written in 1977 and in these times the Western literature wasNEWLINEalmost unknown in the Soviet Union.NEWLINENEWLINEA particularity of this book is that the cones are studied in normed spaces, not always Banach spaces, and some results are more difficult to prove under this condition.NEWLINENEWLINEChapter 3 treats not flattened cones, known in the Western literature as open decomposable cones or \(B\)-cones. There is an interesting example, due to Lozanovskii, Beispiel~4 on page~46, showing that the property of a cone to be closed is essential for the Krein-Shmulian theorem.NEWLINENEWLINEEventually, in Chapter 4, the notion of normal cones is introduced. We mention a nice theorem of Bachtin, Theorem IV.2.2 on page 57: NEWLINELet \((X,K)\) be a Banach space with a closed cone \(K\). If any (o)-bounded increasing sequence of positive elements of \(X\) is norm-bounded, then the cone \(K\) is normal.NEWLINENEWLINE Another important result is Krein's theorem from 1940: For the representation of any functional \(f \in X'\) as a difference of two linear continuous positive functionals, the normality of the cone \(K\) is necessary and sufficient.NEWLINENEWLINEThe Riesz interpolation property is studied in Chapter 5, specifically, Ando's theorem: Let \((X,K)\) an ordered Banach space with a closed and generating cone \(K\). If the dual space \((X', K')\) is a vector lattice, then \((X,K)\) hat the Riesz interpolation property and \(K\) is a normal cone.NEWLINENEWLINERegular and completely regular cones are introduced in Chapter~6. A cone \(K\) in an ordered normed space \((X, K)\) is completely regular if any monotone increasing norm-bounded sequence of elements of \(K\) is a Cauchy sequence. An example of such a cone is given by Theorem VI.4.1 on page~101: A normal cone in a weakly sequentially complete ordered Banach space is completely regular.NEWLINENEWLINEChapters 7--11 treat different properties of cones, such as the paved property, characteristic numbers, the paved property for dual cones. Finally, some results about normed lattices can be found in Chapter~10. Chapter 11 is dedicated to the cone of positive linear operators.NEWLINENEWLINEThe translated version of Vulikh's book contains also some literature and historical complements of theNEWLINERussian edition. These complements are valuable and interesting.NEWLINENEWLINEAs a conclusion, this book by one of the pioneers of the field deserves to be read.