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Does anyone know a book that motivates the beginning of functional analysis in a clear way?

By "clear," I mean that it shows why one would want to define Hilbert spaces and why the theorems are motivated. I know of Dieudonne's History of Functional Analysis, but I am looking for something that also explains the theory.

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    $\begingroup$ Did you try "The prehistory of the theory of distributions" by Jesper Lutzen? $\endgroup$ – Abdelmalek Abdesselam Dec 1 '17 at 21:59
  • $\begingroup$ Lutzen: doi.org/10.1007/978-1-4613-9472-3 $\endgroup$ – David Roberts Dec 1 '17 at 22:17
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    $\begingroup$ I mean that it shows why one would want to define Hilbert spaces and why the theorems are motivated. While you'll probably get plenty of excellent answers to the question as asked, just keep in mind that the actual answer is that nobody had any desire to define Hilbert spaces or to prove theorems for hundreds of thousands of years while the opportunity has always been there. We have many explanations of why one would do this or that after it has been done but all our attempts to project this teleology into the future and to squeeze any predictive value from it fail miserably.. $\endgroup$ – fedja Dec 1 '17 at 22:40
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    $\begingroup$ I highly recommend Pietsch's History of Banach Spaces and Linear Operators, Birkhäuser, 2007. If you know German, the textbook by Dirk Werner contains a wealth of historical information in the corresponding chapters' appendices. $\endgroup$ – Christian Clason Dec 2 '17 at 8:50
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In addition to books on history (J. Dieudonne, History of functional analysis (review) is very good and easy reading), I recommend some books by the founding fathers:

  • John von Neumann, Mathematische Grundlagen der quantum Mechanik, (there are English and Russian translations). Functional analysis (in Hilbert space) is related to quantum mechanics in the same way as Calculus to classical mechanics.

  • F. Riesz and B. Szőkefalvi-Nagy, Functional analysis (multiple editions in French, German, Russian, English).

  • P. Levy, Problèmes concrets d'analyse fonctionnelle. Avec un complément sur les fonctionnelles analytiques par F. Pellegrino. 2d ed. Gauthier-Villars, Paris, 1951. (There is a Russian translation).

  • L. Schwartz, Mathematics for the physical sciences. Hermann, Paris; Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont. 1966.

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    $\begingroup$ I especially second von Neumann's book. This is not how Hilbert spaces first arose, but you would come away from this book completely satisfied about the need to define Hilbert spaces. $\endgroup$ – Nik Weaver Dec 2 '17 at 3:56
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Three articles that I found interesting:

Hellinger, E.; Toeplitz, O., Integralgleichungen und Gleichungen mit unendlichvielen Unbekannten, 184 S. Mit einem Vorwort von E. Hilb. Leipzig, B. G. Teubner (Encyklopädie der mathematischen Wissenschaften mit Einschluß ihrer Anwendungen, II C 13) (1927). ZBL53.0350.01.

Steen, L.A., Highlights in the history of spectral theory, Am. Math. Mon. 80, 359-381 (1973). ZBL0264.46001.

Narici, Lawrence; Beckenstein, Edward, The Hahn-Banach theorem: The life and times, Topology Appl. 77, No.2, 193-211 (1997). ZBL0919.46005.

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