Convex analysis and monotone operator theory in Hilbert spaces.
2nd edition.

*(English)*Zbl 1359.26003
CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC. Cham: Springer (ISBN 978-3-319-48310-8/hbk; 978-3-319-48311-5/ebook). xix, 619 p. (2017).

The first edition of this book appeared in 2011 and was reviewed in [Zbl 1218.47001].

Hédy Attouch wrote with respect to the first edition in the foreword: “Choosing to work in Hilbert spaces offers a wide range of applications, while keeping the mathematics accessible to a large audience. Each topic is developed in a self-contained fashion, and the presentation often draws on recent advances.”

All this is more than ever valid for the new edition which “greatly expands on the first edition, containing over 140 pages of new material, over 270 new results, and more than 100 new exercises. It features a new chapter on proximity operators including two sections on proximity operators of matrix functions, in addition to several new sections distributed throughout the original chapters. Many existing results have been improved, and the list of references has been updated” (cited from the publisher’s description).

Hédy Attouch wrote with respect to the first edition in the foreword: “Choosing to work in Hilbert spaces offers a wide range of applications, while keeping the mathematics accessible to a large audience. Each topic is developed in a self-contained fashion, and the presentation often draws on recent advances.”

All this is more than ever valid for the new edition which “greatly expands on the first edition, containing over 140 pages of new material, over 270 new results, and more than 100 new exercises. It features a new chapter on proximity operators including two sections on proximity operators of matrix functions, in addition to several new sections distributed throughout the original chapters. Many existing results have been improved, and the list of references has been updated” (cited from the publisher’s description).

Reviewer: Bernd Bank (Berlin)

##### MSC:

26-02 | Research exposition (monographs, survey articles) pertaining to real functions |

26B25 | Convexity of real functions of several variables, generalizations |

47-02 | Research exposition (monographs, survey articles) pertaining to operator theory |

47H05 | Monotone operators and generalizations |

49J52 | Nonsmooth analysis |

90C25 | Convex programming |

52A41 | Convex functions and convex programs in convex geometry |

41A65 | Abstract approximation theory (approximation in normed linear spaces and other abstract spaces) |