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?
Best Answer
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.