# Talk:Quasi-uniform convergence

From Encyclopedia of Mathematics

## 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)

- I think you're right: the definition of quasi-uniform continuity is due to Arzela, according to

**How to Cite This Entry:**

Quasi-uniform convergence.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Quasi-uniform_convergence&oldid=42125