fa.functional-analysisho.history-overview
There is a standard trick in analysis, where one chooses a subsequence, then a subsequence of that… and wants to get an eventual subsubsequence of all of them and you take the diagonal. I've always called this the diagonalization trick.
I heard once that this is due to Cantor but haven't been able to find a reference (all searches for diagonal and Cantor lead to his argument about the uncountability of [0,1].
Does anyone have an exact reference?
Thanks.
B.
I think a standard referenz is "Quelques méthodes de résolution des problèmes aux limites non linéaires" from Lions, but it is in french.
Paul Gordan's theses were published in De linea geodetica and digitised by Google, from which I reproduce the relevant page:
Translation:
One can also learn from this publication that the opponents of Gordan at the March 1, 1862 Disputatio were Dr. Kretschmer, Mr. Geiser, and Mr. Rathke, presumably E.E. Kretschmer and C.F. Geiser. Gordan acknowledges as his teachers in mathematics the professors Kummer, Joachimstal, Richelot, Rosenhain, Schroeter, and Galle --- but it is unclear whether any of these were present at the defense ceremony.
Concerning the question "what was actually said at the defense"; at least in my University (Leiden) the practice through the centuries has been that the examination itself is not recorded, the secretary records only the outcome (pass/fail/cum laude etc.).
Best Answer
From this, it sounds like a very early instance is in Ascoli's proof of his theorem: pp. 545-549 of Le curve limite di una varietà data di curve, Atti Accad. Lincei 18 (1884) 521-586. (Which, alas, I can't find online.)
Note that this predates Cantor's argument that you mention (for uncountability of [0,1]) by 7 years.
Edit: I have since found the above-cited article of Ascoli, here. And I must say that the modern diagonal argument is less "obviously there" on pp. 545-549 than Moore made it sound. The notation is different and the crucial subscripts rather hard to read, so at first sight I feel the need for a native Italian speaker to help spot where precisely Ascoli passes to the diagonal sequence (as I guess he must somehow)...
(One may also consult a self-summary of the article in Rend. Reale Istituto lombardo di scienze e lettere 21 (1888) 365-371 here.)