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Derived functors in functional analysis. (English) Zbl 1031.46001

Lecture Notes in Mathematics. 1810. Berlin: Springer. viii, 134 p. EUR 22.95/net; sFr. 39.50; £16.00; $ 29.80 (2003).
The aim of this book is to present in a closed form the homological tools developed recently in functional analysis to treat analytic problems. These problems include for example the surjectivity of linear partial differential operators, the existence of a solution operator for the given operator on spaces of (ultra)differentiable functions, analytic functions or (ultra)distributions, and the continuous and linear extension of analytic or (ultra)differentiable functions. Palamodov in 1969 was the first to realize that several topics in functional analysis could be considered as exactness problems in appropriate categories and thus they could be investigated with the help of derived functors. A major role was played by the projective limit functor. D. Vogt made in the eighties important contributions emphasizing the functional analysis aspects: he investigated the vanishing of the functor \(Ext^1 (E,F)\) for pairs of Fréchet spaces (hence the splitting of exact sequences of Fréchet spaces), and found evaluable conditions to characterize the vanishing of the functor \(Proj^1\) especially for sequence spaces. Vogt discovered also the relation of the vanishing of the functor \(Proj^1\) for a projective spectra of (LB)-spaces with the locally convex properties of the projective limit. His investigations were completed by the work of the author of this volume, Frerick, Braun and others, and found important analytic applications in the work of Domanski, Langenbruch, Meise, Taylor, Vogt and others.
The book consists of seven chapters. The first one is an introduction. The requirements of homological algebra are minimized. In fact, all that is needed in this volume is presented in the nine pages of chapter 2. Chapter 3 develops with detail the theory of the countable projective limit functor, and it constitutes the core of these lecture notes. The results presented here have important consequences and applications in analytic problems. The way the author introduces countable spectra and the projective limit functor differs from the original one of Palamodov. His approach is very simple, but it requires certain arrangements in the applications. These arrangements and the differences to the Palamodov approach are clearly explained. The theorem about the “six spaces” exact sequence of projective limits and derived functors, which is so important in the surjectivity problems for continuous and linear operators, is explained. The vanishing of the functor \(Proj^1\) is an abstract version of the Mittag-Leffler procedure. Three different proofs of Palamodov’s necessary condition for the vanishing of \(Proj^1\) are presented. Each one has its advantages. A complete characterization for projective spectra of Hausdorff (LB)-spaces due to Retakh and Palamodov is explained. Several recent improvements of this result, due to Frerick and the author, and to Braun and Vogt, are included. Vogt’s work about evaluable conditions and the relation with the locally convex properties of the projective limit (e.g., being ultrabornological) are discussed in the interesting section 3.3 about projective limits of locally convex spaces. The special case of projective limits of strong duals of Fréchet Schwartz spaces, here called (LS)-spaces, is carefully analyzed. In the last section 4 of this chapter the author presents several analytic applications of the tools developed in the previous sections. These include the classical Mittag-Leffler theorem, Hörmander’s theorem characterizing the surjective linear partial differential operators \(P(D)\) with constant coefficients on the space of distributions, and the characterization due to Braun, Meise, Taylor, Vogt of the surjectivity of \(P(D)\) on spaces of ultradifferentiable functions of Roumieu type on a convex open set in \(R^n\) in terms of a Phragmen-Lindelöf condition on the zero variety of the polynomial \(P(-z)\). Perhaps the author could have taken the opportunity to mention other applications to the space of real analytic functions, convolution operators or the existence of solution operators or the existence of extension operators.
The author investigates in chapter 4 the homological behaviour of arbitrary projective limits. The author concludes that “the first derived functor for uncountable spectra hardly vanishes and that this theory is much less suitable for analytic applications”. Results in this chapter permit the author to solve in the negative one of the problems of Palamodov in his original article. The first section of chapter 5 introduces the derivatives \(Ext^k(E,.)\) of the functor \(\text{Hom}(E,.)\), and explains the connection to lifting, extension and splitting of exact sequences. If \(X\) is a Fréchet space, there is a close relation between \(\text{Ext}^k(E,X)\) and \(\text{Proj}^kY\) for a suitable spectrum \(Y\). This is used to show that, if \(E\) is a (DF)-space, \(X\) is Fréchet and one of them is nuclear, then \(\text{Ext}^k (E,X)=0\), a result of Palamodov. The rest of the section is joint work with Frerick. The results of chapter 4 are used to disprove a conjecture of Palamodov by showing that \(\text{Ext}^2 (\omega,\varphi)\neq 0\). Wengenroth has continued the investigation of Palamodov’s questions in his article [J. Funct. Anal. 201, 561-571 (2003; Zbl 1038.46063)].
In section 5.2 the author presents Vogt’s theory of the vanishing of \(\text{Ext}^1(E,F)\) for pairs of Fréchet spaces \(E\) and \(F\). The results are obtained mainly as an application of the theorems presented in chapter 3. The completion of Vogt’s arguments by Frerick and Wengenroth is presented. The classical (DN) and \((\Omega)\) splitting theorem of Vogt and Wagner is obtained as a consequence. Its well-known corollaries about the structure of nuclear Fréchet spaces are included. The (PLS)-spaces are precisely the countable projective limits of (LS)-spaces. Section 5.3 presents results due to Domanski and Vogt about the structure of complemented subspaces of the space of distributions \({\mathcal D}'\). The following theorem, which has applications to the splitting of distributional complexes is deduced: Let \(E\) be a (PLS)-space which is isomorphic to a subspace of \({\mathcal D}'\), let \(F\) be a (PLS)-space which is isomorphic to a quotient of \({\mathcal D}'\), then \(\text{Ext}^1 (E,F)=0\) in the category of (PLS)-spaces.
In the sixth chapter Wengenroth explains the relation of acyclic inductive limits of a sequence of Fréchet spaces with the projective limit functor. It is proved that an (LF)-space is acyclic if and only if it satisfies many regularity conditions. This is an important result originally due to the author. The question of Palamodov whether acyclic inductive limits of complete locally convex spaces are acyclic is discussed in this chapter too. If the inductive limit is strict, the answer is positive by a classical result of Köthe. The final chapter treats the problem when the transposed of a homomorphism in the category of locally convex spaces is again a homomorphism. He investigates the duality functors, proves results due to Palamodov, Merzon, S. Dierolf and the reviewer, characterizing quasinormable Fréchet spaces, and explains the essential difficulties which arise beyond the class of Fréchet spaces in connection with this problem.
This is a well-written, useful and informative book. It is reasonably self-contained, and the proofs are neat and complete. It presents the homological tools recently developed in functional analysis to treat analytic problems in the applications of modern locally convex theory to complex analysis, linear partial differential and convolution operators, distribution theory and extensions of analytic or differentiable functions. These lecture notes would be very useful for those interested in the modern theory of Fréchet spaces, (DF)-spaces, general locally convex spaces, and their analytic applications.

MSC:

46-02 Research exposition (monographs, survey articles) pertaining to functional analysis
46M18 Homological methods in functional analysis (exact sequences, right inverses, lifting, etc.)
46A04 Locally convex Fréchet spaces and (DF)-spaces
46M40 Inductive and projective limits in functional analysis

Citations:

Zbl 1038.46063
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