##
**Biorthogonal systems in Banach spaces.**
*(English)*
Zbl 1136.46001

CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC. New York, NY: Springer (ISBN 978-0-387-68914-2/hbk). xviii, 339 p. (2008).

Let \(X\) be a Banach space. A system \((e_\gamma, e_\gamma^*)_{\gamma \in \Gamma} \subset X \times X^*\) is said to be biorthogonal if \(e_\gamma^*(e_\gamma)=1\) and \(e_\gamma^*(e_\alpha)=0\) for \(\alpha \neq \gamma\). If the linear span of \(\{e_\gamma\}_{\gamma \in \Gamma}\) is dense in \(X\), then the system is said to be complete and if a complete biorthogonal system is total (i.e., the linear span of \(\{e_\gamma^*\}_{\gamma \in \Gamma}\) is weak-star dense in \(X^*\)), then it is called an M-basis. Here, “M” comes from Markushevich, who proved in 1943 the existence of an M-basis in every separable Banach space. An M-basis can be considered as a kind of “coordinate system” in \(X\), where the \(e_\gamma\) are the directions of the coordinate axes, and for \(x \in X\) the values of \(e_\gamma^*(x)\) play the role of the corresponding coordinates.

The book under review contains a clear, detailed and self-contained exposition of the modern state-of-the-art in the biorthogonal systems theory. It is written by famous experts, who at the same time are experienced writers of mathematical books.

The nature of the subject is quite different in the separable and the non-separable cases. The separable part (presented in chapters 1 and 2 of the book) is more “classical”, although it contains some sufficiently fresh results, like Terenzi’s theorem on the existence of strong M-bases, and there are old problems which still remain open. The non-separable part is more advanced. In fact, one can consider this monograph as a good introduction to non-separable Banach space theory, because beyond results on specific biorthogonal systems it covers a lot of non-separable techniques, applicable in a very general setting.

The authors stress that one of their goals is to attract young mathematicians to Banach space theory. In my opinion, the book perfectly serves this purpose. While reading this book, one learns a large variety of highly nontrivial ideas and methods, and enjoys beautiful results which introduce the reader into a quickly developing and important mathematical area. For most results, the easiest known proofs are selected, and in some instances substantial simplifications are made. Every chapter contains an exercises section. Exercises (in many cases highly non-trivial) are supplied with hints and references to the corresponding literature. The list of references contains more than 400 items. Let us briefly list the contents of the book.

1. Separable Banach spaces (1.1. Basics; 1.2. Auerbach bases; 1.3. Existence of M-bases in separable spaces; 1.4. Bounded minimal systems; 1.5. Strong M-bases; 1.6. Extensions of M-bases; 1.7. \(\omega\)-independence; 1.8. Exercises).

2. Universality and the Szlenk index (2.1. Trees in Polish spaces; 2.2. Universality for separable spaces; 2.3. Universality of M-bases; 2.4. Szlenk index; 2.5. Szlenk index applications to universality; 2.6. Classification of \(C[0, \alpha]\) spaces; 2.7. Szlenk index and renormings; 2.8. Exercises).

3. Review of weak topology and renormings (3.1. The dual Mackey topology; 3.2. Sequential agreement of topologies in \(X^*\); 3.3. Weak compactness in \(ca(\Sigma)\) and \(L_1(\lambda)\); 3.4. Decompositions of nonseparable Banach spaces; 3.5. Some renorming techniques; 3.6. A quantitative version of Krein’s theorem; 3.7. Exercises).

4. Biorthogonal systems in non-separable spaces (4.1. Long Schauder bases; 4.2. Fundamental biorthogonal systems; 4.3. Uncountable biorthogonal systems in ZFC; 4.4. Nonexistence of uncountable biorthogonal systems; 4.5. Fundamental systems under Martin’s axiom; 4.6. Uncountable Auerbach bases; 4.7. Exercises).

5. Markushevich bases (5.1. Existence of Markushevich bases; 5.2. M-bases with additional properties; 5.3. \(\Sigma\)-subsets of compact spaces; 5.4. WLD Banach spaces and Plichko spaces; 5.5. \(C(K)\) spaces that are WLD; 5.6. Extending M-bases from subspaces; 5.7. Quasicomplements; 5.8. Exercises).

6. Weak compact generating (6.1. Reflexive and WCG Asplund spaces; 6.2. Reflexive generated and Vašák spaces; 6.3. Hilbert generated spaces; 6.4. Strongly reflexive and superreflexive generated spaces; 6.5. Exercises).

7. Transfinite sequence spaces (7.1. Disjointization of measures and applications; 7.2. Banach spaces containing \(\ell_1(\Gamma)\); 7.3. Long unconditional bases; 7.4. Long symmetric bases; 7.5. Exercises).

8. More applications (8.1. Biorthogonal systems and support sets; 8.2. Kunen-Shelah properties in Banach spaces; 8.3. Norm-attaining operators; 8.4. Mazur intersection properties; 8.5. Banach spaces with only trivial isometries; 8.6. Exercises).

The book under review contains a clear, detailed and self-contained exposition of the modern state-of-the-art in the biorthogonal systems theory. It is written by famous experts, who at the same time are experienced writers of mathematical books.

The nature of the subject is quite different in the separable and the non-separable cases. The separable part (presented in chapters 1 and 2 of the book) is more “classical”, although it contains some sufficiently fresh results, like Terenzi’s theorem on the existence of strong M-bases, and there are old problems which still remain open. The non-separable part is more advanced. In fact, one can consider this monograph as a good introduction to non-separable Banach space theory, because beyond results on specific biorthogonal systems it covers a lot of non-separable techniques, applicable in a very general setting.

The authors stress that one of their goals is to attract young mathematicians to Banach space theory. In my opinion, the book perfectly serves this purpose. While reading this book, one learns a large variety of highly nontrivial ideas and methods, and enjoys beautiful results which introduce the reader into a quickly developing and important mathematical area. For most results, the easiest known proofs are selected, and in some instances substantial simplifications are made. Every chapter contains an exercises section. Exercises (in many cases highly non-trivial) are supplied with hints and references to the corresponding literature. The list of references contains more than 400 items. Let us briefly list the contents of the book.

1. Separable Banach spaces (1.1. Basics; 1.2. Auerbach bases; 1.3. Existence of M-bases in separable spaces; 1.4. Bounded minimal systems; 1.5. Strong M-bases; 1.6. Extensions of M-bases; 1.7. \(\omega\)-independence; 1.8. Exercises).

2. Universality and the Szlenk index (2.1. Trees in Polish spaces; 2.2. Universality for separable spaces; 2.3. Universality of M-bases; 2.4. Szlenk index; 2.5. Szlenk index applications to universality; 2.6. Classification of \(C[0, \alpha]\) spaces; 2.7. Szlenk index and renormings; 2.8. Exercises).

3. Review of weak topology and renormings (3.1. The dual Mackey topology; 3.2. Sequential agreement of topologies in \(X^*\); 3.3. Weak compactness in \(ca(\Sigma)\) and \(L_1(\lambda)\); 3.4. Decompositions of nonseparable Banach spaces; 3.5. Some renorming techniques; 3.6. A quantitative version of Krein’s theorem; 3.7. Exercises).

4. Biorthogonal systems in non-separable spaces (4.1. Long Schauder bases; 4.2. Fundamental biorthogonal systems; 4.3. Uncountable biorthogonal systems in ZFC; 4.4. Nonexistence of uncountable biorthogonal systems; 4.5. Fundamental systems under Martin’s axiom; 4.6. Uncountable Auerbach bases; 4.7. Exercises).

5. Markushevich bases (5.1. Existence of Markushevich bases; 5.2. M-bases with additional properties; 5.3. \(\Sigma\)-subsets of compact spaces; 5.4. WLD Banach spaces and Plichko spaces; 5.5. \(C(K)\) spaces that are WLD; 5.6. Extending M-bases from subspaces; 5.7. Quasicomplements; 5.8. Exercises).

6. Weak compact generating (6.1. Reflexive and WCG Asplund spaces; 6.2. Reflexive generated and Vašák spaces; 6.3. Hilbert generated spaces; 6.4. Strongly reflexive and superreflexive generated spaces; 6.5. Exercises).

7. Transfinite sequence spaces (7.1. Disjointization of measures and applications; 7.2. Banach spaces containing \(\ell_1(\Gamma)\); 7.3. Long unconditional bases; 7.4. Long symmetric bases; 7.5. Exercises).

8. More applications (8.1. Biorthogonal systems and support sets; 8.2. Kunen-Shelah properties in Banach spaces; 8.3. Norm-attaining operators; 8.4. Mazur intersection properties; 8.5. Banach spaces with only trivial isometries; 8.6. Exercises).

Reviewer: Vladimir Kadets (Granada)

### MSC:

46-02 | Research exposition (monographs, survey articles) pertaining to functional analysis |

46B15 | Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces |

46B20 | Geometry and structure of normed linear spaces |

46B26 | Nonseparable Banach spaces |

46B03 | Isomorphic theory (including renorming) of Banach spaces |