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Distributions: Topology and Sequential Compactness. May 1, 2014 Contents 1 Introduction 2 2 General definitions, and methods for Topological Vector Spaces. 4 3 The topology of D(Ω) 9 3.1 The spaces DK ..................................... 10 3.2 LF-spaces . 12 3.3 D(Ω) is an LF-space . 14 3.4 Semi-norms defining the topology on D(Ω) . 15 4 Properties of the Space D(Ω) 15 4.1 Completeness of D(Ω) . 15 4.2 Non-metrizability of D(Ω) . 18 4.3 D(Ω) is a Montel Spaces . 19 4.4 Separability . 19 5 The space D0(Ω) 20 5.1 Positive distributions . 21 5.2 Topologies on D0(Ω) . 21 5.3 Non-metrizability . 22 5.4 These Spaces are Hausdorff . 24 5.5 The Banach-Steinhaus Theorem & Barrelled Spaces . 24 5.6 The Banach-Alaoglu-Bourbaki Theorem . 25 5.7 Sequential Compactness . 27 5.8 More Montel Spaces & Equivalence of the Topologies on Bounded sets . 28 5.9 The Weak topology on D(Ω) . 29 5.10 Reflexitivity . 30 5.11 Completeness . 31 5.12 The space D as a subset of D0 and Separability. 32 6 Multiplying Distributions 33 6.1 Multiplication between subspaces of functions and D0 . 33 6.2 The Impossibility of Multiplying two Distributions in General . 34 6.3 Division by Analytic Functions in the Case of One Real Variable. 35 6.3.1 Division by x .................................. 36 6.3.2 Division by Polynomials. 36 1 6.3.3 Division by Analytic Functions with Zeroes of Finite Order. 36 6.4 Applications of Multiplication. 38 7 Rapidly Decreasing Functions. 38 7.1 Relation between S and D(Sn)............................ 40 7.2 The Fourier transform on S ............................. 40 8 Tempered Distributions 41 8.1 Relation between S 0 and D0(Sn)........................... 42 8.2 The Fourier Transform on S '............................. 42 9 The space E0. 43 10 Comparison and Relations between the spaces D; S and E and their duals. 43 11 Tensor Product of two Distributions 44 11.1 The Tensor Product of two Vector Spaces . 44 11.2 The Tensor Product of Function Spaces . 44 11.3 Tensor Products of Distributions . 45 0 11.3.1 The Space L(D(Ω1); D (Ω2)) . 48 11.4 Topological Tensor Products . 48 11.4.1 The π-topology . 48 11.4.2 The -topology . 49 12 Kernels and the kernels theorem. 49 12.1 Regular Kernels . 52 13 Nuclear Locally Convex Spaces 55 13.1 Nuclear Mappings . 55 13.2 Nuclear Spaces . 56 13.3 Nuclear Spaces and the Kernels Theorem. 57 A Appendix: Completions 57 1 Introduction This essay largely concerns the topological properties of the space of distributions as developed by Laurent Schwartz between 1945 and 1950. We begin this introduction with some discussion of the history and uses of distributions and will then move on to an outline of the main topics of the essay. Much of the historical details are taken from [6]. Examples of generalized functions pre-date the rigorous development of distributions. A particularly prevalent example is Heaviside's operational calculus in [12] where he, unrigorously, uses algebraic manipulation of differential operators to treat many problems, particularly in electrodynamics. Schwartz was unaware of Heaviside's calculus when he initially created the theory of distributions. He was told of it by electrical engineers who were able to put the theory of distributions to great use. In [6] L¨utzensays that the theory of distributions was well received by physicists as it \allowed them to use improper functions in good conscience". Though it was in fact Sobolev who first defined distributions as the continuous linear forms on a space of test functions. Schwartz did not know of this work when he began to work on the 2 generalized solutions to partial differential equations. He originally worked by introducing the convolution operators T : D!E which satisfy T (φ ∗ ) = T (φ) ∗ : Here, for example δ is the identity operator. This theory along with many theorems was made by Schwartz in October 1944. It was originally very fruitful and Schwartz was able to prove the analogies of many of the theorems which hold in the current theory of distributions. However, it became more challenging when dealing with Fourier transforms. Consequently, in 1945 Schwartz switched to studying D0. L¨utzenreports in [6] that Schwartz, himself, felt that there were two factors which made D0 a good place to work on his theories of generalized solutions. Firstly, that functional analysis was at that time making great developments and Schwartz's own private work on the dual spaces of Fr´echet spaces allowed him to manipulate these dual spaces without having a `concrete representation'. Secondly, he already knew that δ as a measure could already be represented as a functional. Distribution spaces have many applications, particularly in PDE, which are largely not addressed in this essay, in particular the work done by Lars H¨ormander. Furthermore, the space of tempered distributions allows for many generalizations and extensions of theories using Fourier analysis. The distributions generalize the way in which locally summable functions and measures can act as the kernels of integral operators on an appropriate space of test functions. This, after appropriate checks and with restrictions, allows us to extend the weak formulation of PDEs outside function spaces. This essay begins by introducing the space of test functions and studying its topology, especially results about compactness and metrizability. This space is the smooth functions of compact support. The topology on this space was developed by Schwartz in a method which was extended by Schwartz along with Dieudonn´eto a general class of topological vector space called LF-spaces. We work by first introducing the LF-space in general and seeing how D can be viewed as a particular case. The Theory of Distributions is a theory of duality. We define the space of distributions to be the analytic dual of the test functions. We can also define many operations on the space of distributions; multiplication, differentiation, Fourier transform, in terms of the adjoint map of the maps on the space of test functions where they are defined in the classical sense. We can, therefore, discover many topological properties of the space of distributions by looking at theorems about dual topologies. For example, the Banach-Alaoglu-Bourbaki Theorem which shows us that weak-* bounded, weak-* closed sets in the dual of a Topological Vector Space are weak-* compact. Again in exploring the dual topologies we will focus on compactness and metrizablity results. In fact for both the distributions and the test functions we can show that, although neither space is metrizable, the closed, bounded sets are compact and metrizable (so also sequentially compact). This will lead to the equivalence of the weak-* and strong dual topology on bounded sets and the reflexivity of all spaces. We then explore the multiplication and division of distributions. Multiplication, can be defined by duality between the space E of smooth functions and the distributions. We will explore the continuity properties of this mapping and show that it cannot be extended to a multiplication between two distribution spaces, though here we only look at the case of one variable. We then go on to look at the case where the smooth function we multiply is fixed and examine the invertibility of this operation. This shows another example of how distributions differ from functions. Division by an analytic function is possible in general even when the function has zeros (though these have to be of finite order), the result of division is, however, not uniquely defined. We finish the section on multiplication by very briefly showing how these 3 results make clear and rigorous how to pose linear PDE with smooth coefficients in the space of distributions. We will go on to look at some other spaces of distributions which are found by enlarging the space of test functions. These are the space S 0 of tempered distributions which is defined as the dual of S the space of rapidly decreasing function. This space is particularly useful for Fourier analysis, which we will look at extremely briefly. We also look at the space E0 which are the distributions of compact support. The topology of all these spaces are very similar and we will not reprove all the results that we gave for the space D0. The next section looks at tensor products of distributions which allows us to develop the theory of kernel operators and examines the way in which an element of D0(Ω × Ω) can act as a mapping from D(Ω) to D0(Ω). We also look at the topology on tensor products as developed by Grothendiek in an example of how the theory of distributions fed back into functional analysis. We will demonstrate Schwartz's Kernels Theorem which gives that all linear mappings form D(Ω) to D0(Ω) can be given in this way. This was proved by Schwartz for example in [11] or [10] using the theory of nuclear locally convex spaces which was again work by Grothendieck when he was Schwartz's student. This theory is very beautiful but in this section we show how the algebraic, not the topological isomorphism, can be found from a much simpler proof using Fourier transforms. We then look at fundamental kernels and parametrices for differential operators and we relate their regularity properties to the hypo-ellipticity of the operator. We finish by briefly giving some notions of Nuclear spaces including the `idea' of Schwartz's proof of the kernels theorem and of a very elegant similar proof given in [4].
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