Talk:Renato Caccioppoli

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I'm wondering if someone can reconstruct what was meant by the following passage:

In 1932 he introduced the general concept of inversion of functional correspondence, showing that a transformation between two Banach spaces is invertible only if it is locally invertible and if the only convergent sequences are the compact ones.

A convergent sequence not including its limit point is not compact in the relative topology (in a Hausdorff space), whereas if we extrapolate it to "precompact", the condition trivializes. Could I have mistranslated? Here's the Italian text:

una trasformazione tra due spazi di Banach è invertibile solo se è invertibile localmente e se le uniche a divenire successioni convergenti sono le successioni compatte.

It should be possible to figure out what was meant, but I haven't yet. --Trovatore 16:03, 4 March 2006 (UTC)[reply]

Trovatore, your translation is correct. It is a statement of the famous Hadamard-Caccioppoli theorem: Jacques Hadamard proved the finite dimensional version, while Renato Caccioppoli extended it to the infinite-dimensional Banach space setting. You can find a clear statement and a proof of it in monographs on the fixed point theory, like those by James Dugundji and Andrzej Granas, or the one of Vasile Istratescu. The whole entry need a rework and a clean up, and I'll do it in the future: however, I need to collect more informations on Caccioppoli, since many authors wrote about him but, due to peculiarities of his personality and to his political convictions, not all the things written about him are effectively reliable. Some of them made a caricature of him, emphasizing the most unusual aspects of his temperament. Daniele.tampieri (talk) 09:57, 9 April 2011 (UTC)[reply]

After translating this article, I took a look at this biography listed in the refs, and it appears that the Italian source material must have been based on it quite closely. (Or, I suppose, just possibly the other way around.) Does anyone understand just how much you have to rewrite text like this to make it not a copyvio? --Trovatore 16:08, 4 March 2006 (UTC)[reply]

Hi, according to this page, "Caccioppoli proved the analyticity of the solutions of the elliptical equations of class C2, solving the nineteenth Hilbert problem". However, the nineteenth problem had been solved by Sergei Natanovich Bernstein in 1904 (see wikipedia pages devoted to Hilbert's 19th problem and Bernstein).

Cheers —The preceding unsigned comment was added by 140.105.16.64 (talk • contribs) 09:59, 5 October 2006 (UTC)