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Part II. Duality. Spaces of Distributions
P A R T I1
Duality. Spaces of Distributions
In this part, the reader will find an exposition of the main body of distribution theory and of the theory of duality between topological b', 9') are defined vector spaces. The main spaces of distributions (9, as duals of spaces of 3 ' ?" functions (of '3?7,grn, and 9, respectively). The standard operations- differentiation, multiplication by a V" function, convolution, Fourier transformation- are systematically defined as transposes of analog operations in spaces of V" functions. I t is evident that such an approach, by duality and transposition, requires a minimum amount of knowledge about these concepts. This is provided in Chapter 18 (The Theorem of Hahn-Banach), 19 (Topologies on the Dual), and 23 (Transpose of a Continuous Linear Map). In the Chapter presenting the Hahn-Banach theorem, a few pages are devoted to showing how the theorem is used in the treatment of various problems (e.g., problems of approximation, of existence of solutions to a functional equation, and also problems of separation of convex sets). I n Chapters 20, 21, and 22, examples of duals are given; Chapter 20 is entirely devoted to the duality between Lp and Lp', the so-called Lebesgue spaces (and also between l p and P I , the spaces of sequences). A proof of Holder's inequality is given. Chapter 21 studies the dual of the space of continuous functions with compact support, which is the space of Radon measures, then the dual of the space of Vmfunctions with compact support, which is the space of distributions. Chapter 22 studies two cases of duality of a somewhat more abstract nature: the duality between polynomials and formal power series, and the duality between entire analytic functions in C" and analytic functionals. We prove the important theorem that the Fourier-Bore1 transformation is an isomorphism of the space of analytic functionals onto the space of entire functions of exponential type in C" (this theorem may be viewed as describing the duality between entire functions and entire functions of exponential type; this duality is closely related to the one between polynomials and power series). At the end of Chapters 20, 21, and 22, we find ourselves with a stock of spaces in duality that should provide us with a good number of examples on which to rely in the later study of duality. I t should be pointed out, however, that we have at our disposal the space of all the distributions, 9,but that we are not yet able to identify any one of its subspaces to the duals of the other spaces of functions which have 111
178
DUALITY
been introduced. In order to be able to do this, we need the notion of transpose of a linear map and the fact that the transpose is injective whenever the image of the map is dense. For then we may take advantage of the fact that the natural injection of V; into L p (1 p < +m), Vt, V k(0 k +a),and 9 has a dense image. Consequently, the dual of each one of these spaces can be identified with a linear subspace of 9, i.e., can be regarded as a space of distributions. The notion of transpose, needless to say, is important in many respects, the material treated in Chapters 23-38 bears witness to this. I t is by transposition that we define the (linear partial) differential operators acting on distributions (Chapter 23), the Fourier transformation of tempered distributions (Chapter 25), and the convolution of distributions (Chapter 27). Transposition is the key to the study of the weak dual topology, as carried through in Chapter 35 (where attention is centered on the dual of a subspace and the dual of a quotient space and the related weak topologies), and to the study of reflexivity (Chapter 36: in the terms set down by Mackey and Bourbaki, with particular emphasis on reflexive Banach spaces, on one hand, and on Monte1 spaces, on the other). The main theorem in Chapter 37, due to S. Banach, may be regarded as the culmination of this line of thought: it shows the equivalence between the surjectivity of a continuous linear map of a FrCchet space into another FrCchet space, and the property that its transpose be one-to-one and have a weakly closed image. This theorem is complemented with a characterization of weakly closed linear subspaces in the dual of a FrCchet space, also due to Banach. In order to impress the importance of these theorems on the mind of the student, Chapter 38 (the last in Part 11) shows how they can be applied to the proof of a classical theorem of E. Bore1 and also to the proof of one of the main results about existence of V" solutions of linear partial differential equations (this last application is essentially due to B. Malgrange). We have preceded these chapters by a description of the standard aspects of distribution theory: the support of a distribution is introduced in Chapter 24, where the main theorem of structure is stated and proved; the procedures of approximation of distributions by cutting and regularizing are described in Chapter 28; the Fourier transforms of distributions with compact support are characterized in Chapter 29 (this characterization forms the celebrated Paley-Wiener theorem). In Chapter 30, we show that Fourier transformation exchanges, SO to speak, multiplication and convolution. We have added a section (Chapter 26) on convolution of functions, where we prove the Minkowski-Hblder-Young inequality. We have thought that it was appropriate to add also a rather lengthy section (Chapter 31) on
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SPACES OF DISTRIBUTIONS
179
Sobolev’s spaces: these spaces play an increasingly important role in the theory of linear (and even of nonlinear) partial differential equations, and it is mainly with the application of functional analysis to partial differential equations in mind that the material presented here has been selected. Finally, no exposition of the theory of topological vector spaces, even admittedly succinct, could dispense with the statement and the proof of the Banach-Steinhaus theorem; we fulfill this obligation in Chapter 33. We give some of the applications of the theorem in Chapter 34. As it is a statement about equicontinuous sets of linear maps, we introduce these sets in Chapter 32 and establish their main properties. The Banach-Steinhaus theorem is extensively applied in the section on reflexivity (Chapter 36).
Duality. Spaces of Distributions
In this part, the reader will find an exposition of the main body of distribution theory and of the theory of duality between topological b', 9') are defined vector spaces. The main spaces of distributions (9, as duals of spaces of 3 ' ?" functions (of '3?7,grn, and 9, respectively). The standard operations- differentiation, multiplication by a V" function, convolution, Fourier transformation- are systematically defined as transposes of analog operations in spaces of V" functions. I t is evident that such an approach, by duality and transposition, requires a minimum amount of knowledge about these concepts. This is provided in Chapter 18 (The Theorem of Hahn-Banach), 19 (Topologies on the Dual), and 23 (Transpose of a Continuous Linear Map). In the Chapter presenting the Hahn-Banach theorem, a few pages are devoted to showing how the theorem is used in the treatment of various problems (e.g., problems of approximation, of existence of solutions to a functional equation, and also problems of separation of convex sets). I n Chapters 20, 21, and 22, examples of duals are given; Chapter 20 is entirely devoted to the duality between Lp and Lp', the so-called Lebesgue spaces (and also between l p and P I , the spaces of sequences). A proof of Holder's inequality is given. Chapter 21 studies the dual of the space of continuous functions with compact support, which is the space of Radon measures, then the dual of the space of Vmfunctions with compact support, which is the space of distributions. Chapter 22 studies two cases of duality of a somewhat more abstract nature: the duality between polynomials and formal power series, and the duality between entire analytic functions in C" and analytic functionals. We prove the important theorem that the Fourier-Bore1 transformation is an isomorphism of the space of analytic functionals onto the space of entire functions of exponential type in C" (this theorem may be viewed as describing the duality between entire functions and entire functions of exponential type; this duality is closely related to the one between polynomials and power series). At the end of Chapters 20, 21, and 22, we find ourselves with a stock of spaces in duality that should provide us with a good number of examples on which to rely in the later study of duality. I t should be pointed out, however, that we have at our disposal the space of all the distributions, 9,but that we are not yet able to identify any one of its subspaces to the duals of the other spaces of functions which have 111
178
DUALITY
been introduced. In order to be able to do this, we need the notion of transpose of a linear map and the fact that the transpose is injective whenever the image of the map is dense. For then we may take advantage of the fact that the natural injection of V; into L p (1 p < +m), Vt, V k(0 k +a),and 9 has a dense image. Consequently, the dual of each one of these spaces can be identified with a linear subspace of 9, i.e., can be regarded as a space of distributions. The notion of transpose, needless to say, is important in many respects, the material treated in Chapters 23-38 bears witness to this. I t is by transposition that we define the (linear partial) differential operators acting on distributions (Chapter 23), the Fourier transformation of tempered distributions (Chapter 25), and the convolution of distributions (Chapter 27). Transposition is the key to the study of the weak dual topology, as carried through in Chapter 35 (where attention is centered on the dual of a subspace and the dual of a quotient space and the related weak topologies), and to the study of reflexivity (Chapter 36: in the terms set down by Mackey and Bourbaki, with particular emphasis on reflexive Banach spaces, on one hand, and on Monte1 spaces, on the other). The main theorem in Chapter 37, due to S. Banach, may be regarded as the culmination of this line of thought: it shows the equivalence between the surjectivity of a continuous linear map of a FrCchet space into another FrCchet space, and the property that its transpose be one-to-one and have a weakly closed image. This theorem is complemented with a characterization of weakly closed linear subspaces in the dual of a FrCchet space, also due to Banach. In order to impress the importance of these theorems on the mind of the student, Chapter 38 (the last in Part 11) shows how they can be applied to the proof of a classical theorem of E. Bore1 and also to the proof of one of the main results about existence of V" solutions of linear partial differential equations (this last application is essentially due to B. Malgrange). We have preceded these chapters by a description of the standard aspects of distribution theory: the support of a distribution is introduced in Chapter 24, where the main theorem of structure is stated and proved; the procedures of approximation of distributions by cutting and regularizing are described in Chapter 28; the Fourier transforms of distributions with compact support are characterized in Chapter 29 (this characterization forms the celebrated Paley-Wiener theorem). In Chapter 30, we show that Fourier transformation exchanges, SO to speak, multiplication and convolution. We have added a section (Chapter 26) on convolution of functions, where we prove the Minkowski-Hblder-Young inequality. We have thought that it was appropriate to add also a rather lengthy section (Chapter 31) on
< <
<
SPACES OF DISTRIBUTIONS
179
Sobolev’s spaces: these spaces play an increasingly important role in the theory of linear (and even of nonlinear) partial differential equations, and it is mainly with the application of functional analysis to partial differential equations in mind that the material presented here has been selected. Finally, no exposition of the theory of topological vector spaces, even admittedly succinct, could dispense with the statement and the proof of the Banach-Steinhaus theorem; we fulfill this obligation in Chapter 33. We give some of the applications of the theorem in Chapter 34. As it is a statement about equicontinuous sets of linear maps, we introduce these sets in Chapter 32 and establish their main properties. The Banach-Steinhaus theorem is extensively applied in the section on reflexivity (Chapter 36).