##
**Locally convex spaces.**
*(English)*
Zbl 1287.46002

Graduate Texts in Mathematics 269. Cham: Springer (ISBN 978-3-319-02044-0/hbk; 978-3-319-02045-7/ebook). viii, 213 p. (2014).

According to the author’s introduction, the aim of this book is to cover most of the general theory of locally convex spaces that is needed for applications to other areas of analysis, since for most practicing analysts the restriction to Banach spaces is not enough. The material in the book is based on a course that the author taught at the University of Washington, covering the first five chapters. The author intends to present a broad enough range of spaces keeping the material within reasonable bounds. As he explains, his presentation owes much to Rudin’s classical book on functional analysis, although duality is only treated there for Banach spaces.

Chapter 1 reviews concepts of point set topology and topological groups. Chapter 2 presents topological vector spaces, (absolutely) convex sets, bounded sets and linear operators. Locally convex spaces, seminorms and the Hahn-Banach theorem are explained at the beginning of Chapter 3. It also includes a study of duality: dual spaces, the bipolar theorem and topologies of uniform convergence. Fréchet and (LF)-spaces are treated in the last two sections. The author only considers what are usually called strict (LF)-spaces. The proofs of the properties of strict (LF)-spaces are rather long and cumbersome. There are better presentations in classical texts on locally convex spaces. The reader interested in a modern presentation of (general) (LF)-spaces and (LB)-spaces, their relation to homological methods in functional analysis, and their applications to analytic problems, should have a look at the lecture notes by J. Wengenroth [Derived functors in functional analysis. Berlin: Springer (2003; Zbl 1031.46001)]. Chapter 4 studies barrelled and bornological spaces, the uniform boundedness theorem, the open mapping and the closed graph theorem. Chapter 5 starts with the transpose of a linear map, investigates reflexive and Montel locally convex spaces, proves the Krein-Milman theorem, presents the Ptak closed graph theorem, the closed range theorem, the Riesz-Leray theorem, and applications to spectral theory of compact operators between locally convex spaces. De Wilde’s closed graph and open mapping theorems, which are very useful in applications, are not mentioned. The short Chapter 6 includes a few results about duals of Fréchet spaces, including the Banach-Dieudonné theorem and some properties of (DF)-spaces.

Each chapter finishes with a set of exercises. There are four appendices. The first one contains complements on point set topology, such as the Urysohn metrization theorem or the Tychonoff theorem about the product of compact sets. The second one explains the closed graph theorem for homomorphisms between topological groups. The third one includes Eberlein’s and Krein-Smulian’s theorems. The last one collects hints for selected exercises.

Dover Publications has recently re-edited the classical books on the theory of locally convex topological vector spaces by J. Horvath [Topological vector spaces and distributions. Vol. I (1966; Zbl 0143.15101)], F. Trèves [Topological vector spaces, distributions and kernels. New York-London: Academic Press (1967; Zbl 0171.10402)] and A. Wilansky [Modern methods in topological vector spaces. Düsseldorf etc.: McGraw-Hill (1978; Zbl 0395.46001)], which are mentioned in the bibliography. They constitute an excellent complement to the text under review. They contain many more results, examples and analytic applications, but they cover much more material than can be explained in a one year course. The booklet of A. P. Robertson and W. Robertson [Topological vector spaces. Cambridge University Press (1964; Zbl 0123.30202)], which is not in the references list, covers material similar to the book under review and is still worth looking at.

The book is well written, it is easy to read and should be useful for a one semester course. The proofs are clear and easy to follow and there are many exercises. The book presents in an accessible way the classical theory of locally convex spaces, and can be useful especially for beginners interested in different areas of analysis, who are familiar with elementary point set topology, linear algebra and the Banach space theory taught in a beginning real analysis course. There is not much originality in the presentation or in the examples and not many applications are included, but this monograph is still a good addition to the literature on this topic.

The reader interested in the modern theory of locally convex spaces, including Fréchet and (DF)-spaces, nuclear spaces, Köthe echelon spaces, the isomorphic classification of subspaces and quotients of power series spaces, the splitting of short exact sequences and the applications to complex analysis, Schwartz distribution theory and linear partial differential operator, should look carefully at Part IV of the book by R. Meise and D. Vogt [Introduction to functional analysis. Transl. from the German by M. S. Ramanujan. Oxford: Clarendon Press (1997; Zbl 0924.46002)].

Chapter 1 reviews concepts of point set topology and topological groups. Chapter 2 presents topological vector spaces, (absolutely) convex sets, bounded sets and linear operators. Locally convex spaces, seminorms and the Hahn-Banach theorem are explained at the beginning of Chapter 3. It also includes a study of duality: dual spaces, the bipolar theorem and topologies of uniform convergence. Fréchet and (LF)-spaces are treated in the last two sections. The author only considers what are usually called strict (LF)-spaces. The proofs of the properties of strict (LF)-spaces are rather long and cumbersome. There are better presentations in classical texts on locally convex spaces. The reader interested in a modern presentation of (general) (LF)-spaces and (LB)-spaces, their relation to homological methods in functional analysis, and their applications to analytic problems, should have a look at the lecture notes by J. Wengenroth [Derived functors in functional analysis. Berlin: Springer (2003; Zbl 1031.46001)]. Chapter 4 studies barrelled and bornological spaces, the uniform boundedness theorem, the open mapping and the closed graph theorem. Chapter 5 starts with the transpose of a linear map, investigates reflexive and Montel locally convex spaces, proves the Krein-Milman theorem, presents the Ptak closed graph theorem, the closed range theorem, the Riesz-Leray theorem, and applications to spectral theory of compact operators between locally convex spaces. De Wilde’s closed graph and open mapping theorems, which are very useful in applications, are not mentioned. The short Chapter 6 includes a few results about duals of Fréchet spaces, including the Banach-Dieudonné theorem and some properties of (DF)-spaces.

Each chapter finishes with a set of exercises. There are four appendices. The first one contains complements on point set topology, such as the Urysohn metrization theorem or the Tychonoff theorem about the product of compact sets. The second one explains the closed graph theorem for homomorphisms between topological groups. The third one includes Eberlein’s and Krein-Smulian’s theorems. The last one collects hints for selected exercises.

Dover Publications has recently re-edited the classical books on the theory of locally convex topological vector spaces by J. Horvath [Topological vector spaces and distributions. Vol. I (1966; Zbl 0143.15101)], F. Trèves [Topological vector spaces, distributions and kernels. New York-London: Academic Press (1967; Zbl 0171.10402)] and A. Wilansky [Modern methods in topological vector spaces. Düsseldorf etc.: McGraw-Hill (1978; Zbl 0395.46001)], which are mentioned in the bibliography. They constitute an excellent complement to the text under review. They contain many more results, examples and analytic applications, but they cover much more material than can be explained in a one year course. The booklet of A. P. Robertson and W. Robertson [Topological vector spaces. Cambridge University Press (1964; Zbl 0123.30202)], which is not in the references list, covers material similar to the book under review and is still worth looking at.

The book is well written, it is easy to read and should be useful for a one semester course. The proofs are clear and easy to follow and there are many exercises. The book presents in an accessible way the classical theory of locally convex spaces, and can be useful especially for beginners interested in different areas of analysis, who are familiar with elementary point set topology, linear algebra and the Banach space theory taught in a beginning real analysis course. There is not much originality in the presentation or in the examples and not many applications are included, but this monograph is still a good addition to the literature on this topic.

The reader interested in the modern theory of locally convex spaces, including Fréchet and (DF)-spaces, nuclear spaces, Köthe echelon spaces, the isomorphic classification of subspaces and quotients of power series spaces, the splitting of short exact sequences and the applications to complex analysis, Schwartz distribution theory and linear partial differential operator, should look carefully at Part IV of the book by R. Meise and D. Vogt [Introduction to functional analysis. Transl. from the German by M. S. Ramanujan. Oxford: Clarendon Press (1997; Zbl 0924.46002)].

Reviewer: José Bonet (Valencia)

### MSC:

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

46A03 | General theory of locally convex spaces |

46A04 | Locally convex Fréchet spaces and (DF)-spaces |

46A08 | Barrelled spaces, bornological spaces |

46A13 | Spaces defined by inductive or projective limits (LB, LF, etc.) |

46A30 | Open mapping and closed graph theorems; completeness (including \(B\)-, \(B_r\)-completeness) |