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In which paper/book (most likely) by either Sobolev or Schwartz is the Dirac function properly and explicitly substantiated as a functional (tempered distribution), preferably quoting Dirac's name? I need to know the oldest (the original) source, in order to quote it as a remark in an article that I'm writing on the history of mathematical foundations of Dirac's inventions (Delta function and bra-kets).

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    $\begingroup$ Schwartz L., Généralisation de la notion de fonction et de dérivation théorie des distributions, Annales Des Télécommunications vol. 3, p.135–140 (1948): a plausible place (online Springer) $\endgroup$
    – sand1
    Apr 29 at 8:57
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    $\begingroup$ Sobolev had papers in 1936 and 1938 in which he introduced "generalized functions" and also proved his imbedding theorems. Surely that did include Dirac's function... but the Math Reviews accounts (unsurprisingly) do not talk about the papers from that viewpoint. $\endgroup$ Apr 29 at 16:22
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I found another, later paper which also references the Schwartz item that @sand1 commented as well as another paper by Schwartz from 1945. The reference paper is:

Historia Mathematics 10 (1983) 149-183 DISTRIBUTIONS: THE EVOLUTION OF A MATHEMATICAL THEORY BY JOHN SYNOWIEC INDIANA UNIVERSITY NORTHWEST, GARY, INDIANA 46408

The references listed therein include

Schwartz, L. 1944. Sur certaines families non fondamentales de fonctions continues. Bulletin de la Société Mathematique de France 72, 141-145.

    1. Généralisation de la notion de fonction, de dérivation, de transformation de Fourier et applications mathematiques et physiques. Annales de l'Université de Grenoble 21, 57-74. -- 1947-1948. Theorie des distributions et transformations de Fourier. Annales de l'université de Grenoble 23, 7-24.
    1. Généralisation de la notion de fonction et de dérivation. Théorie de distributions. Annales Télécommunications 3, 135-140.

and many more.

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  • $\begingroup$ I accepted this answer as the one I was looking for since the article of 1945 in AUG is the first one treating Dirac's 1926 invention as a functional in an explicit way. $\endgroup$
    – DanielC
    Apr 29 at 23:26

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