##
**Elements of mathematics. Topological vector spaces. Chapters 1–5.
(Éléments de mathématique. Espaces vectoriels topologiques. Chapitres 1 à 5.)**
*(French)*
Zbl 0482.46001

Paris etc.: Masson. vii, 368 p. (1981).

This is the hardbound “new” edition of Book V of Bourbaki’s treatise. The first two chapters were published originally in 1953 (Zbl 0050.10703) and an extensive revision came out in 1966 (Zbl 0145.37702). The man change compared to the second edition is that the appendix to Chapter II concerning the Markoff-Kakutani fixed point theorem has been considerably extended, e.g., with the Ryll-Nardzewski theorem about the existence of a G-invariant point, and appended to Chapter IV. There are a few major additions (e.g., the existence of a lifting for linear maps of Fréchet spaces), many smaller ones and some stylistic improvements.

Banach’s homomorphism theorem is now stated in the following form: Let \(E\) and \(F\) be two metrizable topological vector spaces over a non-discrete valued field, assume \(E\) complete, and let \(u: E\to F\) be a continuous linear map; then the following conditions are equivalent: (i) \(u\) is a surjective strict morphism, (ii) \(F\) is complete and \(u\) surjective, (iii) \(\operatorname{Im}(u)\) is not meager in \(F\), (iv) for every neighborhood \(V\) of \(0\) in \(E\), \(\overline{u(V)}\) is a neighborhood of \(0\) in \(F\) (i.e., \(u\) is nearly open in Pták’s terminology, whose theory, however, is not mentioned in the book, not even in the exercises).

An example of the small additions is the sentence “Posons \(y_0 = g(x_0)\)” on page 1.10, line 3, since the symbol \(y_0\) went undefined in the preceding edition (unfortunately \(y\) appears in the text instead of \(y_0\)).

About 15 new exercises have been added, e.g., paracomplete spaces, Fuchssteiner’s and Edwards’ sandwich theorems and Heinz Onig’s results on sublinear functions.

To the best of the reviewer’s knowledge, no new edition of Chapters III–V appeared since the first one in 1955 (Zbl 0066.35301), so these chapters have been “entirely recast”. Chapter III now begins with the concept of a bornology in a topological vector space, attributed generally to Waelbroeck and popularized by Buchwalter and mainly by Hogbe-Nlend. As new examples we have the spaces of infinitely differentiable functions of distribution theory, Gevrey spaces and spaces of holomorphic functions. The barrelled spaces, which started the chapter in the first edition, have been moved to where they belong: the section on the Banach-Steinhaus theorem. §2 is also new: bornological spaces have been promoted from the exercises to the main text. The old §3 on the spaces \(\mathcal L_{\mathfrak G}(E;F)\) has been cut into two: the new §3, which kept the title of the old one: “Espaces d’applications linéaires continues”, and §4 on the Banach-Steinhaus theorem. Grothendieck’s completion theorem, with the nice proof first found in the book of Kelley and Namioka, is now in the text of §3 instead of the exercises, which makes it possible to deduce from it Banach’s theorem characterizing weak*-continuous linear forms on the dual of a complete space, and to replace the old proof by the natural one.

Sequentially complete spaces are introduced in the text (under the name “semi-complet”) and in some statements they replace the quasi-complete ones of the earlier edition, e.g., in the theorem which gives conditions that simply bounded subsets of \(\mathcal L(E;F)\) shall be bounded for some \(\mathfrak G\)-topology.

The reviewer regrets that Theorem 3 of the old §3, no. 6 concerning the equicontinuity of a set of maps \((x,t)\mapsto f(x,t)\), where \(f\) is linear in \(x\) only, has been eliminated: it was useful for instance to extend properties of bilinear maps to \(n\)-linear ones by induction; its replacement (Proposition 2 of §5, no. 2) assumes that \(f\) is linear in both variables.

There are several new exercises, e.g., on the barrelledness of countable-codimensional subspaces of a barrelled space and Valdivia’s example of a bornological, barrelled space which is not ultrabornological. §5 on hypocontinuous bilinear maps is almost unchanged except for the addition just mentioned and that equihypocontinuous sets have been relegated to the exercises.

A new section contains L. Schwartz’s Borel graph theorem, Lusinian locally convex spaces and Douady’s result on the continuity of universally measurable linear maps. The exercises develop De Wilde’s theory of the closed graph theorem for spaces with absolutely convex webs of type C, called here “exhaustions”.

The first section of the old Chapter IV on weak topologies has been moved to Chapter II already in the second edition. Chapter IV has been augmented considerably (though the main text is still only 46 pages long): the topologies of subspaces, quotient spaces, products and sums are examined in much more detail, Dieudonné’s theorem concerning the second countability of metrizable Montel spaces is presented, there are newly included results on Fréchet spaces and on strict morphisms between them, sequentially barrelled (“semi-tonnelé”) spaces are defined, it is proved that the strong duals of Fréchet spaces are (DF)-spaces (though the term itself is introduced only in exercise 2), and there is a subsection on criteria of surjectivity. The new §5 contains, among others, the theorem of Eberlein and Šrnulian (formerly in exercises) and Krein’s theorem on the compactness of the balanced, closed, convex hull of a compact set (the proof uses Lebesgue’s bounded convergence theorem on a compact space). Some old exercises have been moved in the main text and there are about three dozen of new ones, e.g., on Fredholm operators, Schauder bases (“base banachique”) and the space of R. C. James. At the end of Chapter IV there are two useful tables, one on the interdependence of the principal types of locally convex spaces, and one on the principal bornologies on the dual of a locally convex space. The old §5 on the duality of Banach spaces has been omitted and is replaced by a list at the end of the book which summarizes some important properties of Banach spaces.

The original two sections of Chapter V (elementary theory of Hilbert spaces) are almost unchanged except for a few new examples (e.g., Sobolev spaces, the Hardy space \(H^2(D))\) and some new terminology (like “semi-norme préhilbertienne”, a hermitian form is “séparante” instead of “non-dégénérée”, orthogonal projectors are now also called “orthoprojecteurs”), but two very welcome sections have been added to the chapter which has thus increased to more than twice of its original length. The first new section is about tensor products and symmetric and exterior powers of Hilbert spaces, which have gained such importance in e.g., infinite-dimensional holomorphy or quantum theory; both the symmetric and the antisymmetric Fock spaces and the Wick product are listed as examples. The last section discusses the adjoint of a linear map and several classes of operators in Hilbert spaces: automorphisms (unitary operators), partially isometric, normal, hermitian, positive, Hilbert-Schmidt operators, and operators with finite trace. The bibliography has been augmented by the book of Lindenstrauss and Tzafriri, Grothendieck’s thesis and his São Paulo notes.

When the first edition of this book was published, it was, except for Grothendieck’s Sao Paulo notes, the only treatment of the theory of locally convex spaces which emerged a few years earlier from the works of Köthe, Mackey, Dieudonné, Schwartz, Grothendieck and others. Since then a number of monographs and textbooks have been written on the subject by Köthe, Kelley-Namioka et al., A. P. and W. Robertson, H. Schaefer, Wilansky, Jarchow, to name just a few. Of course Bourbaki’s goal is different from that of the authors of these books; he writes not for the intended specialist but presents only “what every mathematician should know”, at least in an ideal world. One then understands that he does not want various types of spaces to proliferate, but with uncharacteristic timidity he mentions by name three types of spaces in the exercises only, instead of the main text where they would have found natural places:

1) Infrabarrelled spaces for which the canonical injection into the bidual is a strict morphism. Thus in Chap. IV, §1, no. 2, Proposition 4 he states that \(E\) has the Mackey topology if \(E\) is barrelled or bornological, when the natural hypothesis would be that \(E\) is infrabarrelled; the same is true about Chap. IV, §2, no. 1, Proposition 2.

2) Ultrabornological spaces which have a large role in closed graph theorems, they figure for instance implicitly in the Borel graph theorem Chap. III, p. 34. In Chap. IV, §3, no. 4, Proposition 4 and its Corollary the true conclusion is that the strong dual of a reflexive Fréchet space is ultrabornological. Similarly on page II.37 just before no. 7, instead of assuming that \(E\) is bornological and complete, it is more natural to assume that \(E\) is ultrabornological.

3) Distinguished spaces which figure implicitly in Proposition 3 of Chapter IV, §2, no. 1. Theorem 2 of Chapter IV, §3, no. 6 is true for distinguished spaces and not only for reflexive Fréchet spaces.

Some time ago there was a rumor that topological tensor products and nuclear spaces will figure in the new edition of the book. Later it was said that these topics will be placed into Théories Spectrales, and this seems to be confirmed by a reference to TS,V in the example on page IV.25, and again on page V.50 it is said that nuclear maps will be defined later. It is to be hoped that a place will be found also for Schwartz spaces which have remarkable stability properties, are frequently used and are not mentioned even in the exercises of the present volume.

This edition depends more than the preceding ones on the Book of General Topology, in particular on Chapters II (topological groups) and X (function spaces).

There are some slips and unusually many misprints, some of them quite disturbing:

p.II.2, line 5 from below: for I, p.47 read I, p.7; p.II.9, line 13: for \(\alpha(x)\ge 0\) read \(g(x)\ge 0\); p.II.16, line 5 from below: for prop. 16 read prop. 17; p.II.30, lines 11 and 12 from below: the same line is repeated; p.II.55, line 7 from below: for profit read produit; p.II.58, line 5 from below: for \(h(z)\ge \) read \(h(z)\le \); p.II.69, line 6 from below: for \(C\cap D \ne\emptyset\) read \(C\cap D' \ne\emptyset\); p.II.78, exercise 8: no reference is given to the place where “linearly compact” is defined (Alg. Comm. Chap. III, §2, exercise 15); p.II.94, line 5 from below: omit the second “extrémal”; p.II.95, line 3 from below: the notation \(C_A\) is used without any reference (p.II.71, exerc. 14);

p.III.8, last line: for \(f(x)\in F\) read \(\hat f(x)\in F\); p.III.42, line 14: for \(a\in L\) read \(a\notin L\);

p.IV.15: Proposition 3 is repeated as exercise 4b on page IV.52; p. IV.45, line 8: for \(C(X;\mathbb R)\) read \(C(G;\mathbb R)\); p.IV.54, line 8: for \(E''\subset E\) read \(E''\subset \hat E\); p.IV.74, line 15: for \(\beta(S)\) read \(\mathcal B(S)\);

p.V.2., immediately after the formula \(f(x,y)= \sum_{j,k}\alpha_{ij}\bar \xi_j\eta_k\) on line 5, the following has been left out: ”en posant \(\alpha_{jk} = f(e_j,e_k)\); en outre l’identité (3) équivaut à la condition \(\alpha_{jk} = \bar\alpha_{kj}\)”; p.V.45: the first line is missing and the second line is repeated; p.V.46, line 13 from below: for \(\mathcal H(E)\) read \(\mathcal F\); p.V.63: exercise 15 e) uses Hilbert sums introduced only in the next section; p.V.75, line 2 from below: for “sur \(N\)” read “sur \(\tilde N\)”.

Though Bourbaki’s treatise is meant to be a textbook, it is also used as a reference, therefore it is helpful if the results listed in the exercises are labeled with the names traditionally attached to them; this is done in most cases but e.g., Chap. I, §2, exerc. 15 is Auerbach’s lemma, Chap. II, §5, exerc. 15 is Kirchberger’s theorem and exerc. 27 is a minimax theorem.

Finally, a few concepts introduced in the exercises (gliding hump: IV, p.54, conorm: V, p.75, Ptolemaic inequality: V. p.60, paracomplete: I, p.29, etc.) are not listed in the terminological index.

Banach’s homomorphism theorem is now stated in the following form: Let \(E\) and \(F\) be two metrizable topological vector spaces over a non-discrete valued field, assume \(E\) complete, and let \(u: E\to F\) be a continuous linear map; then the following conditions are equivalent: (i) \(u\) is a surjective strict morphism, (ii) \(F\) is complete and \(u\) surjective, (iii) \(\operatorname{Im}(u)\) is not meager in \(F\), (iv) for every neighborhood \(V\) of \(0\) in \(E\), \(\overline{u(V)}\) is a neighborhood of \(0\) in \(F\) (i.e., \(u\) is nearly open in Pták’s terminology, whose theory, however, is not mentioned in the book, not even in the exercises).

An example of the small additions is the sentence “Posons \(y_0 = g(x_0)\)” on page 1.10, line 3, since the symbol \(y_0\) went undefined in the preceding edition (unfortunately \(y\) appears in the text instead of \(y_0\)).

About 15 new exercises have been added, e.g., paracomplete spaces, Fuchssteiner’s and Edwards’ sandwich theorems and Heinz Onig’s results on sublinear functions.

To the best of the reviewer’s knowledge, no new edition of Chapters III–V appeared since the first one in 1955 (Zbl 0066.35301), so these chapters have been “entirely recast”. Chapter III now begins with the concept of a bornology in a topological vector space, attributed generally to Waelbroeck and popularized by Buchwalter and mainly by Hogbe-Nlend. As new examples we have the spaces of infinitely differentiable functions of distribution theory, Gevrey spaces and spaces of holomorphic functions. The barrelled spaces, which started the chapter in the first edition, have been moved to where they belong: the section on the Banach-Steinhaus theorem. §2 is also new: bornological spaces have been promoted from the exercises to the main text. The old §3 on the spaces \(\mathcal L_{\mathfrak G}(E;F)\) has been cut into two: the new §3, which kept the title of the old one: “Espaces d’applications linéaires continues”, and §4 on the Banach-Steinhaus theorem. Grothendieck’s completion theorem, with the nice proof first found in the book of Kelley and Namioka, is now in the text of §3 instead of the exercises, which makes it possible to deduce from it Banach’s theorem characterizing weak*-continuous linear forms on the dual of a complete space, and to replace the old proof by the natural one.

Sequentially complete spaces are introduced in the text (under the name “semi-complet”) and in some statements they replace the quasi-complete ones of the earlier edition, e.g., in the theorem which gives conditions that simply bounded subsets of \(\mathcal L(E;F)\) shall be bounded for some \(\mathfrak G\)-topology.

The reviewer regrets that Theorem 3 of the old §3, no. 6 concerning the equicontinuity of a set of maps \((x,t)\mapsto f(x,t)\), where \(f\) is linear in \(x\) only, has been eliminated: it was useful for instance to extend properties of bilinear maps to \(n\)-linear ones by induction; its replacement (Proposition 2 of §5, no. 2) assumes that \(f\) is linear in both variables.

There are several new exercises, e.g., on the barrelledness of countable-codimensional subspaces of a barrelled space and Valdivia’s example of a bornological, barrelled space which is not ultrabornological. §5 on hypocontinuous bilinear maps is almost unchanged except for the addition just mentioned and that equihypocontinuous sets have been relegated to the exercises.

A new section contains L. Schwartz’s Borel graph theorem, Lusinian locally convex spaces and Douady’s result on the continuity of universally measurable linear maps. The exercises develop De Wilde’s theory of the closed graph theorem for spaces with absolutely convex webs of type C, called here “exhaustions”.

The first section of the old Chapter IV on weak topologies has been moved to Chapter II already in the second edition. Chapter IV has been augmented considerably (though the main text is still only 46 pages long): the topologies of subspaces, quotient spaces, products and sums are examined in much more detail, Dieudonné’s theorem concerning the second countability of metrizable Montel spaces is presented, there are newly included results on Fréchet spaces and on strict morphisms between them, sequentially barrelled (“semi-tonnelé”) spaces are defined, it is proved that the strong duals of Fréchet spaces are (DF)-spaces (though the term itself is introduced only in exercise 2), and there is a subsection on criteria of surjectivity. The new §5 contains, among others, the theorem of Eberlein and Šrnulian (formerly in exercises) and Krein’s theorem on the compactness of the balanced, closed, convex hull of a compact set (the proof uses Lebesgue’s bounded convergence theorem on a compact space). Some old exercises have been moved in the main text and there are about three dozen of new ones, e.g., on Fredholm operators, Schauder bases (“base banachique”) and the space of R. C. James. At the end of Chapter IV there are two useful tables, one on the interdependence of the principal types of locally convex spaces, and one on the principal bornologies on the dual of a locally convex space. The old §5 on the duality of Banach spaces has been omitted and is replaced by a list at the end of the book which summarizes some important properties of Banach spaces.

The original two sections of Chapter V (elementary theory of Hilbert spaces) are almost unchanged except for a few new examples (e.g., Sobolev spaces, the Hardy space \(H^2(D))\) and some new terminology (like “semi-norme préhilbertienne”, a hermitian form is “séparante” instead of “non-dégénérée”, orthogonal projectors are now also called “orthoprojecteurs”), but two very welcome sections have been added to the chapter which has thus increased to more than twice of its original length. The first new section is about tensor products and symmetric and exterior powers of Hilbert spaces, which have gained such importance in e.g., infinite-dimensional holomorphy or quantum theory; both the symmetric and the antisymmetric Fock spaces and the Wick product are listed as examples. The last section discusses the adjoint of a linear map and several classes of operators in Hilbert spaces: automorphisms (unitary operators), partially isometric, normal, hermitian, positive, Hilbert-Schmidt operators, and operators with finite trace. The bibliography has been augmented by the book of Lindenstrauss and Tzafriri, Grothendieck’s thesis and his São Paulo notes.

When the first edition of this book was published, it was, except for Grothendieck’s Sao Paulo notes, the only treatment of the theory of locally convex spaces which emerged a few years earlier from the works of Köthe, Mackey, Dieudonné, Schwartz, Grothendieck and others. Since then a number of monographs and textbooks have been written on the subject by Köthe, Kelley-Namioka et al., A. P. and W. Robertson, H. Schaefer, Wilansky, Jarchow, to name just a few. Of course Bourbaki’s goal is different from that of the authors of these books; he writes not for the intended specialist but presents only “what every mathematician should know”, at least in an ideal world. One then understands that he does not want various types of spaces to proliferate, but with uncharacteristic timidity he mentions by name three types of spaces in the exercises only, instead of the main text where they would have found natural places:

1) Infrabarrelled spaces for which the canonical injection into the bidual is a strict morphism. Thus in Chap. IV, §1, no. 2, Proposition 4 he states that \(E\) has the Mackey topology if \(E\) is barrelled or bornological, when the natural hypothesis would be that \(E\) is infrabarrelled; the same is true about Chap. IV, §2, no. 1, Proposition 2.

2) Ultrabornological spaces which have a large role in closed graph theorems, they figure for instance implicitly in the Borel graph theorem Chap. III, p. 34. In Chap. IV, §3, no. 4, Proposition 4 and its Corollary the true conclusion is that the strong dual of a reflexive Fréchet space is ultrabornological. Similarly on page II.37 just before no. 7, instead of assuming that \(E\) is bornological and complete, it is more natural to assume that \(E\) is ultrabornological.

3) Distinguished spaces which figure implicitly in Proposition 3 of Chapter IV, §2, no. 1. Theorem 2 of Chapter IV, §3, no. 6 is true for distinguished spaces and not only for reflexive Fréchet spaces.

Some time ago there was a rumor that topological tensor products and nuclear spaces will figure in the new edition of the book. Later it was said that these topics will be placed into Théories Spectrales, and this seems to be confirmed by a reference to TS,V in the example on page IV.25, and again on page V.50 it is said that nuclear maps will be defined later. It is to be hoped that a place will be found also for Schwartz spaces which have remarkable stability properties, are frequently used and are not mentioned even in the exercises of the present volume.

This edition depends more than the preceding ones on the Book of General Topology, in particular on Chapters II (topological groups) and X (function spaces).

There are some slips and unusually many misprints, some of them quite disturbing:

p.II.2, line 5 from below: for I, p.47 read I, p.7; p.II.9, line 13: for \(\alpha(x)\ge 0\) read \(g(x)\ge 0\); p.II.16, line 5 from below: for prop. 16 read prop. 17; p.II.30, lines 11 and 12 from below: the same line is repeated; p.II.55, line 7 from below: for profit read produit; p.II.58, line 5 from below: for \(h(z)\ge \) read \(h(z)\le \); p.II.69, line 6 from below: for \(C\cap D \ne\emptyset\) read \(C\cap D' \ne\emptyset\); p.II.78, exercise 8: no reference is given to the place where “linearly compact” is defined (Alg. Comm. Chap. III, §2, exercise 15); p.II.94, line 5 from below: omit the second “extrémal”; p.II.95, line 3 from below: the notation \(C_A\) is used without any reference (p.II.71, exerc. 14);

p.III.8, last line: for \(f(x)\in F\) read \(\hat f(x)\in F\); p.III.42, line 14: for \(a\in L\) read \(a\notin L\);

p.IV.15: Proposition 3 is repeated as exercise 4b on page IV.52; p. IV.45, line 8: for \(C(X;\mathbb R)\) read \(C(G;\mathbb R)\); p.IV.54, line 8: for \(E''\subset E\) read \(E''\subset \hat E\); p.IV.74, line 15: for \(\beta(S)\) read \(\mathcal B(S)\);

p.V.2., immediately after the formula \(f(x,y)= \sum_{j,k}\alpha_{ij}\bar \xi_j\eta_k\) on line 5, the following has been left out: ”en posant \(\alpha_{jk} = f(e_j,e_k)\); en outre l’identité (3) équivaut à la condition \(\alpha_{jk} = \bar\alpha_{kj}\)”; p.V.45: the first line is missing and the second line is repeated; p.V.46, line 13 from below: for \(\mathcal H(E)\) read \(\mathcal F\); p.V.63: exercise 15 e) uses Hilbert sums introduced only in the next section; p.V.75, line 2 from below: for “sur \(N\)” read “sur \(\tilde N\)”.

Though Bourbaki’s treatise is meant to be a textbook, it is also used as a reference, therefore it is helpful if the results listed in the exercises are labeled with the names traditionally attached to them; this is done in most cases but e.g., Chap. I, §2, exerc. 15 is Auerbach’s lemma, Chap. II, §5, exerc. 15 is Kirchberger’s theorem and exerc. 27 is a minimax theorem.

Finally, a few concepts introduced in the exercises (gliding hump: IV, p.54, conorm: V, p.75, Ptolemaic inequality: V. p.60, paracomplete: I, p.29, etc.) are not listed in the terminological index.

Reviewer: John Michael Horváth (College Park)

### MSC:

46Axx | Topological linear spaces and related structures |

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

46Exx | Linear function spaces and their duals |

46C05 | Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product) |

46M05 | Tensor products in functional analysis |