Vector duality and its applications. (Vektornaya dvojstvennost’ i ee prilozheniya).

*(Russian)*Zbl 0616.49010
Novosibirsk: Izdatel’stvo ”Nauka” Sibirskoe Otdelenie. 256 p. R. 1.80 (GĂ¶ 86 A 3178) (1985).

The book is a monograph devoted to vector duality in spaces of linear operators.

It consists of five chapters. In the first one the method of cyclic compactness is exposed. By means of it some basic facts of the classical theory of duality are extended to vector bilinear forms that take on their values in Kantorovitch spaces. In the next chapter the general position method is considered. According to the algebraic equivalents of the Hahn-Banach-Kantorovitch theorems, topological existence theorems are obtained. A method to obtain Boolean-valued realisations of such vector duality objects as K-spaces and Banach-Kantorovitch spaces can be found in the third chapter.

Some applications of the theory of vector duality are given in the last two parts. New results for analytic representations of linear operators are obtained. In the fifth chapter some important techniques for extremal problem analysis such as Fenchel’s transform, \(\epsilon\)-subdifferential and subdifferential are discussed.

It consists of five chapters. In the first one the method of cyclic compactness is exposed. By means of it some basic facts of the classical theory of duality are extended to vector bilinear forms that take on their values in Kantorovitch spaces. In the next chapter the general position method is considered. According to the algebraic equivalents of the Hahn-Banach-Kantorovitch theorems, topological existence theorems are obtained. A method to obtain Boolean-valued realisations of such vector duality objects as K-spaces and Banach-Kantorovitch spaces can be found in the third chapter.

Some applications of the theory of vector duality are given in the last two parts. New results for analytic representations of linear operators are obtained. In the fifth chapter some important techniques for extremal problem analysis such as Fenchel’s transform, \(\epsilon\)-subdifferential and subdifferential are discussed.

Reviewer: M.Todorov

##### MSC:

49N15 | Duality theory (optimization) |

46A20 | Duality theory for topological vector spaces |

49-02 | Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control |

47A67 | Representation theory of linear operators |

49J45 | Methods involving semicontinuity and convergence; relaxation |

49J52 | Nonsmooth analysis |