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Difference between revisions of "Talk:Quasi-uniform convergence"

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(→‎Name of theorem: several Arzelà-Ascoli theorems?)
(→‎Name of theorem: Apparently the definition of quasi-uniform continuity is indeed due to Arzela)
 
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:"In 1883 Arzelà (1983/1984) [2] gave a necessary and sufficient condition via quasi-uniform convergence for the pointwise limit of a sequence of real-valued continuous functions on a compact interval to be continuous. Arzelà's work paved the way for several outstanding papers. A milestone was the P.S. Alexandroff convergence introduced in 1948 to tackle the question for a sequence of continuous functions from a topological space (not necessarily compact) to a metric space. In 2009, in the realm of metric spaces, Beer and Levi (2009) [10] found another necessary and sufficient condition through the novel notion of strong uniform convergence on finite sets. We offer a direct proof of the equivalence of Arzelà, Alexandroff and Beer–Levi conditions." [http://www.sciencedirect.com/science/article/pii/S0022247X10004531 Agata Casertaa, Giuseppe Di Maio, L˘ubica Holá 2010]. [[User:Boris Tsirelson|Boris Tsirelson]] ([[User talk:Boris Tsirelson|talk]]) 23:32, 18 October 2017 (CEST)
 
:"In 1883 Arzelà (1983/1984) [2] gave a necessary and sufficient condition via quasi-uniform convergence for the pointwise limit of a sequence of real-valued continuous functions on a compact interval to be continuous. Arzelà's work paved the way for several outstanding papers. A milestone was the P.S. Alexandroff convergence introduced in 1948 to tackle the question for a sequence of continuous functions from a topological space (not necessarily compact) to a metric space. In 2009, in the realm of metric spaces, Beer and Levi (2009) [10] found another necessary and sufficient condition through the novel notion of strong uniform convergence on finite sets. We offer a direct proof of the equivalence of Arzelà, Alexandroff and Beer–Levi conditions." [http://www.sciencedirect.com/science/article/pii/S0022247X10004531 Agata Casertaa, Giuseppe Di Maio, L˘ubica Holá 2010]. [[User:Boris Tsirelson|Boris Tsirelson]] ([[User talk:Boris Tsirelson|talk]]) 23:32, 18 October 2017 (CEST)
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::I think you're right: the definition of quasi-uniform continuity is due to Arzela, according to ''Scenes from the History of Real Functions'', F.A. Medvedev p.104 [https://books.google.co.uk/books?id=rqmABwAAQBAJ&pg=PA104].  The statement of the article [[Arzelà-Ascoli theorem]] needs correction.  [[User:Richard Pinch|Richard Pinch]] ([[User talk:Richard Pinch|talk]]) 19:12, 19 October 2017 (CEST)

Latest revision as of 17:12, 19 October 2017

Name of theorem

There seems to be almost no support for the name "Arzelà–Aleksandrov theorem" as opposed to "Arzelà-Ascoli theorem" which is what it is called elsewhere in EOM. Richard Pinch (talk) 20:08, 18 October 2017 (CEST)

Really? I only know this Arzelà-Ascoli theorem. But maybe there are several Arzelà-Ascoli theorems? Boris Tsirelson (talk) 22:36, 18 October 2017 (CEST)
Even in this text entitled "Arzela-Ascoli theoremS" I did not find anything like that... Boris Tsirelson (talk) 22:48, 18 October 2017 (CEST)
And still, Russian version says the same as our English version. Strange. Boris Tsirelson (talk) 22:53, 18 October 2017 (CEST)
In GILLESPIE AND HURWITZ 1930, "Arzela condition" is mentioned on pages 528 and 538; otherwise Arzela is not mentioned. Boris Tsirelson (talk) 23:06, 18 October 2017 (CEST)
In Wolff 1919 it is called Arzela's theorem. Boris Tsirelson (talk) 23:27, 18 October 2017 (CEST)
"In 1883 Arzelà (1983/1984) [2] gave a necessary and sufficient condition via quasi-uniform convergence for the pointwise limit of a sequence of real-valued continuous functions on a compact interval to be continuous. Arzelà's work paved the way for several outstanding papers. A milestone was the P.S. Alexandroff convergence introduced in 1948 to tackle the question for a sequence of continuous functions from a topological space (not necessarily compact) to a metric space. In 2009, in the realm of metric spaces, Beer and Levi (2009) [10] found another necessary and sufficient condition through the novel notion of strong uniform convergence on finite sets. We offer a direct proof of the equivalence of Arzelà, Alexandroff and Beer–Levi conditions." Agata Casertaa, Giuseppe Di Maio, L˘ubica Holá 2010. Boris Tsirelson (talk) 23:32, 18 October 2017 (CEST)
I think you're right: the definition of quasi-uniform continuity is due to Arzela, according to Scenes from the History of Real Functions, F.A. Medvedev p.104 [1]. The statement of the article Arzelà-Ascoli theorem needs correction. Richard Pinch (talk) 19:12, 19 October 2017 (CEST)
How to Cite This Entry:
Quasi-uniform convergence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quasi-uniform_convergence&oldid=42125