Hake's Theorem, due to Heinrich Hake of Düsseldorf in 1921, says that an improper Henstock–Kurzweil integral (aka generalized Riemann integral, gauge integral, Perron integral, or Denjoy integral) on a bounded interval is already proper. That is, if $f$ is defined on a half-open interval $[a,c)$, $f$ is HK-integrable on $[a,b]$ for each $b$ satisfying $a \leq b < c$, and the limit of $\int_a^b f$ as $b \to c^-$ exists, then we may define $f(c)$ however we like and find that $f$ is integrable on $[a,c]$ and that $\int_a^c f$ equals the aforementioned limit. (The converse is also true; if $\int_a^c f$ exists, then it may be calculated as a limit.)

I can't find anything about this Hake. Most references just say ‘Hake's theorem’, a few say ‘H. Hake’, and Bartle's book on the HK integral says ‘Heinrich Hake in 1921’. Bartle also gives a reference, an article in Mathematische Annalen, whose byline says ‘Heinrich Hake in Düsseldorf’. And that's it.

Various online search attempts give me references to this theorem, the contemporary Düsseldorf telephone directory, and the 18th-century law professor Ludolf Heinrich Hake, but nothing more about the 20th-century mathematician who proved the theorem. Does anybody know anything about him?

  • $\begingroup$ I see that the tag (biography) was created in this question. Tags with only one occurrence are removed after 6 months unless they have tag-wiki. So perhaps you could create some basic tag-info if you think that this tag is worth keeping. (Feel free to ping me either here or in chat after you read this comment so that I know that it is no longer needed and I can delete it.) $\endgroup$ Aug 12, 2018 at 5:30
  • $\begingroup$ @MartinSleziak : I don't remember creating a tag, but I probably also didn't pay attention to how much something was used if it showed up when I started typing; is it possible that something was using this tag and the tag was later removed? Anyway, I don't really want to get into tag management and advocacy. It would be nice to have a tag more specific than #ho.history-overview for biographies of mathematicians, but I'll leave that up to others to decide. $\endgroup$ Aug 12, 2018 at 20:04
  • $\begingroup$ I'd guess if we want to continue this discussion, it would be better to do so somewhere else - so that we do not leave here too many comments unrelated to the question (perhaps in chat). But probably the easiest solution is to leave the tag to its fate - as you suggested. $\endgroup$ Aug 12, 2018 at 20:40

1 Answer 1


The Deutsche Mathematiker-Vereinigung has a biographical entry: Heinrich Hake was born August 4, 1891 in Düsseldorf. He attended high school in Attendorn (Westfalen), studied in Göttingen and Bonn, where he completed his studies in 1914. After military service he conducted Ph.D. research with prof. Hans Hahn in Bonn, where he defended his dissertation on 2 March 1921, entitled: Über de la Vallée Poussins Ober- und Unterfunktionen einfacher Integrale und die Integraldefinition von Perron. It was published in Math. Ann. 83 (1921) 119-148.

No date of death is listed, but it must have been after 1945.

I note that Bonn University marked the 125 birthday of Heinrich Hake in a memorial calendar.

  • $\begingroup$ If I may ask, why do you infer that his death was after 1945? Thanks... $\endgroup$ Jan 30, 2018 at 1:00
  • $\begingroup$ According to Zentralblatt (which is rather complete for German math) he had only one publication. So probably he did something else, not mathematics, after his PhD. $\endgroup$ Jan 30, 2018 at 2:31
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    $\begingroup$ @paul garret: The biography entry link given by Carlo says that Hanke was a member of the German Math. Soc. from 1921 to circa 1945. $\endgroup$
    – F Zaldivar
    Jan 30, 2018 at 3:15
  • $\begingroup$ Ah, thanks, @FZaldivar. (I am evidently too lazy!) $\endgroup$ Jan 30, 2018 at 3:57
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    $\begingroup$ The linked biography contains the statement "StR Düsseldorf noch 1942, OStDir Leibniz-G Düsseldorf 1.1.46", Here "StR" is the abbreviation for "Studienrat", a typical teacher pay grade, and "OStDir" is an abbreviation for "Oberstudiendirektor", the usual pay grade for head teachers. It seems he became a teacher after his PhD, still was one in 1942, and became head teacher of the Leibniz-Gymnasium grammar school in Düsseldorf in 1946. $\endgroup$
    – Kusma
    Feb 5, 2018 at 10:11

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