##
**Problemi di esistenza in analisi funzionale.**
*(Italian)*
Zbl 0041.43401

Sc. Norm. Super. Pisa. Quaderni Mat. 3, Pisa: Litografia Tacchi. 184 p. (1949).

In his preface, the author remarks that great advance has been made recently in the theory of differential and partial differential equations. In this connection he mentions, on the one side, the topological methods due to Schauder and Leray and, on the other, Caccioppoli’s methods of inversion of certain functional operators. The volume under review gives the substance of lectures delivered at two Italian Universities and intends to serve as an introduction and a guide to Research students. The author warns them from the outset that this is not a systematic treatise. The time has not yet come to write it. But a monograph surveying a wealth of new methods and results was certainly a much felt need and the author must be warmly congratulated to have successfully met it.

A first chapter is devoted to the theory of topological, metrical, spaces, specially of the Banach spaces. The fundamental notions and theorems are clearly and concisely recalled, although the advance made in this field since the classical book of Banach is a little neglected: for instance, no mention is made of important memoirs by Dieudonné, Mackey, Schwartz; also the extremely important class of Banach spaces studied in great detail by Schauder, in particular in the article referred to under No. 66, is only casually utilized in the second chapter. A proof due to Banach himself of the so-called Hahn-Banach theorem concludes the first chapter, no mention being made of the very original proof given by J. Dieudonné [see Rev. Sci. 79, 642–643 (1941; JFM 67.0404.03; Zbl 0063.01104)].

The greatest part of the second chapter, perhaps the most useful of the book for young Research students, deals with the problem of inversion of certain functional transformations. The first four paragraphs are about basic notions. Then the theory of the inversion of linear transformations is summarily set forth, followed by an extremely elegant application to the classical Dirichlet problem which is due to the author. The third and fourth paragraphs would have gained much had some reference been made to the article of J. Dieudonné [Ann. Sci. Éc. Norm. Supér., III. Ser. 59, 107–139 (1942; Zbl 0027.32101)]. The rest of the chapter is concerned with Caccioppoli’s methods. These methods are based on results established “locally” and yield, under certain hypotheses, “global” theorems.

Among the many questions raised in this chapter and which remain open, that of finding whether the conditions given under II and II’, (pages 78 and 88) – where an essential part is played by sequences – are not too artificial and cannot be broadened seems particularly important to the reviewer. As it has been shown, in particular by A. Weil, great progress is always achieved when one rids topological theories of countability conditions.

Numerous examples illustrate Caccioppoli’s theory, including an important one, due to the author, page 110. It would be interesting to compare carefully Caccioppoli’s and Schauder’s methods. It looks as if every one of the problems solved by the former might be solved also by the latter; but there may be cases in which the former have definite advantages.

No reference is made to the work of G. Giraud. It is true that Giraud never used topological methods and stuck to the classical ones: generalized potentials, integral equations, successive approximations. Yet his results in the theory of linear equations of the elliptic type have an extreme importance in the applications of topological methods to the theory of nonlinear equations of the same type.

Speaking of omissions, there is little doubt that students of topological methods would have wished that the limitations of either Caccioppoli’s, or Schauder’s, or Leray’s methods should have been carefully scrutinized. In some cases unwelcomed restrictions are needed – for instance assumptions about the “smallness” of a domain – and the knowledge of a certain number of awkward limitations may serve as a powerful incentive to further investigation. The author might have mentioned also that very little is known about the applications of topological methods to systems of partial differential equations.

The third chapter is a rapid survey of the topological methods of Schauder and Leray. The sketchy summary of the basic notions of Algebraic Topology might have been avoided, had the book under review followed and not preceded by a little more than a few months the remarkable “Introduction to Topology” of S. Lefschetz [Princeton, N. J.: Princeton University Press (1949; Zbl 0041.51801)]; the same remark applies equally well to what is said about Brouwer’s degree of a transformation. But the author should have at least mentioned the very important article of J. Leray [J. Math. Pur. Appl. (9) 24, 201–248 (1946; Zbl 0060.40705)].

The fixed point theorem of Brouwer is dealt with only in the last paragraph of the third chapter. No reference is made to the most elegant proof of it given by W. Hurewicz and H. Wallman, in their “Dimension Theory”, [Princeton, N. J.: Princeton University Press (1948; Zbl 0036.12501), page 40].

Schauder’s extension of this celebrated theorem to Banach spaces is only summarized and one should have liked it to be treated at greater length, with a number of applications. The author does not mention that Schauder succeeded in extending Brouwer’s theorem to linear, complete, metrical spaces. Most likely a striking application of the fixed point theorem to equations of the parabolic type is not known to the author having been, unfortunately, published by S. Minakshisundaram in a rather inaccessible Journal [J. Madras Univ., Sect. B 14, 73–142 (1942; Zbl 0061.22102)].

In analysing the very readable and interesting monograph the reviewer may have insisted much on some criticisms and not given sufficient praise to the very clear and suggestive presentation of a difficult but most attractive subject; nor perhaps has he laid enough stress on the precious bibliography, at the end of the book, in which 108 items are to be found. It would serve a useful purpose if an enlarged and printed edition of this lithographed monograph could be soon made available to a large mathematical public.

A first chapter is devoted to the theory of topological, metrical, spaces, specially of the Banach spaces. The fundamental notions and theorems are clearly and concisely recalled, although the advance made in this field since the classical book of Banach is a little neglected: for instance, no mention is made of important memoirs by Dieudonné, Mackey, Schwartz; also the extremely important class of Banach spaces studied in great detail by Schauder, in particular in the article referred to under No. 66, is only casually utilized in the second chapter. A proof due to Banach himself of the so-called Hahn-Banach theorem concludes the first chapter, no mention being made of the very original proof given by J. Dieudonné [see Rev. Sci. 79, 642–643 (1941; JFM 67.0404.03; Zbl 0063.01104)].

The greatest part of the second chapter, perhaps the most useful of the book for young Research students, deals with the problem of inversion of certain functional transformations. The first four paragraphs are about basic notions. Then the theory of the inversion of linear transformations is summarily set forth, followed by an extremely elegant application to the classical Dirichlet problem which is due to the author. The third and fourth paragraphs would have gained much had some reference been made to the article of J. Dieudonné [Ann. Sci. Éc. Norm. Supér., III. Ser. 59, 107–139 (1942; Zbl 0027.32101)]. The rest of the chapter is concerned with Caccioppoli’s methods. These methods are based on results established “locally” and yield, under certain hypotheses, “global” theorems.

Among the many questions raised in this chapter and which remain open, that of finding whether the conditions given under II and II’, (pages 78 and 88) – where an essential part is played by sequences – are not too artificial and cannot be broadened seems particularly important to the reviewer. As it has been shown, in particular by A. Weil, great progress is always achieved when one rids topological theories of countability conditions.

Numerous examples illustrate Caccioppoli’s theory, including an important one, due to the author, page 110. It would be interesting to compare carefully Caccioppoli’s and Schauder’s methods. It looks as if every one of the problems solved by the former might be solved also by the latter; but there may be cases in which the former have definite advantages.

No reference is made to the work of G. Giraud. It is true that Giraud never used topological methods and stuck to the classical ones: generalized potentials, integral equations, successive approximations. Yet his results in the theory of linear equations of the elliptic type have an extreme importance in the applications of topological methods to the theory of nonlinear equations of the same type.

Speaking of omissions, there is little doubt that students of topological methods would have wished that the limitations of either Caccioppoli’s, or Schauder’s, or Leray’s methods should have been carefully scrutinized. In some cases unwelcomed restrictions are needed – for instance assumptions about the “smallness” of a domain – and the knowledge of a certain number of awkward limitations may serve as a powerful incentive to further investigation. The author might have mentioned also that very little is known about the applications of topological methods to systems of partial differential equations.

The third chapter is a rapid survey of the topological methods of Schauder and Leray. The sketchy summary of the basic notions of Algebraic Topology might have been avoided, had the book under review followed and not preceded by a little more than a few months the remarkable “Introduction to Topology” of S. Lefschetz [Princeton, N. J.: Princeton University Press (1949; Zbl 0041.51801)]; the same remark applies equally well to what is said about Brouwer’s degree of a transformation. But the author should have at least mentioned the very important article of J. Leray [J. Math. Pur. Appl. (9) 24, 201–248 (1946; Zbl 0060.40705)].

The fixed point theorem of Brouwer is dealt with only in the last paragraph of the third chapter. No reference is made to the most elegant proof of it given by W. Hurewicz and H. Wallman, in their “Dimension Theory”, [Princeton, N. J.: Princeton University Press (1948; Zbl 0036.12501), page 40].

Schauder’s extension of this celebrated theorem to Banach spaces is only summarized and one should have liked it to be treated at greater length, with a number of applications. The author does not mention that Schauder succeeded in extending Brouwer’s theorem to linear, complete, metrical spaces. Most likely a striking application of the fixed point theorem to equations of the parabolic type is not known to the author having been, unfortunately, published by S. Minakshisundaram in a rather inaccessible Journal [J. Madras Univ., Sect. B 14, 73–142 (1942; Zbl 0061.22102)].

In analysing the very readable and interesting monograph the reviewer may have insisted much on some criticisms and not given sufficient praise to the very clear and suggestive presentation of a difficult but most attractive subject; nor perhaps has he laid enough stress on the precious bibliography, at the end of the book, in which 108 items are to be found. It would serve a useful purpose if an enlarged and printed edition of this lithographed monograph could be soon made available to a large mathematical public.

Reviewer: Charles Racine (Madras)

### MSC:

46-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to functional analysis |