As indicated in the introduction one must work consistently with smooth “test functions” in the theory of distributions. In this chapter we have collected the basic facts that one needs to know about such functions. As an introduction a brief summary of differential calculus is given in Section 1.1. It is written with a reader in mind who has seen the material before but perhaps with different emphasis and different notation. The reader who finds the presentation hard to follow is recommended to study first a more thorough modern treatment of differential calculus in several variables, and experienced readers should proceed directly to Section 1.2. In addition to the basic indispensible facts we have included in Sections 1.3 and 1.4 some more refined constructions which will be useful some time in this book but are not important for the main theme. The reader in a hurry may therefore wish to omit Section 1.3 from Theorem 1.3.5 on and also Theorem 1.4.2, Lemma 1.4.3 and the rest of Section 1.4 from Theorem 1.4.6 on.

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Authors and Affiliations

1. Department of Mathematics, University of Lund, Box 118, SE-22100, Lund, Sweden

   Lars Hörmander

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