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 matheamatician who proved the theorem. Does anybody know anything about him?
✓ Extra quality
ExtraProxies brings the best proxy quality for you with our private and reliable proxies
✓ Extra anonymity
Top level of anonymity and 100% safe proxies – this is what you get with every proxy package
✓ Extra speed
1,ooo mb/s proxy servers speed – we are way better than others – just enjoy our proxies!
USA proxy location
We offer premium quality USA private proxies – the most essential proxies you can ever want from USA
99,9% servers uptime
No usage restrictions
Perfect for SEO
We are working 24/7 to bring the best proxy experience for you – we are glad to help and assist you!