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Preliminaries
Published in Hugo D. Junghenn, Principles of Analysis, 2018
We have seen that every closed ball in Rd $ \mathbb R ^{d} $ is compact. By contrast, closed balls in the space C[0, 1] with the supremum norm are not compact, as may be inferred from the fact that the sequence of functions fn(x)=xn $ f_n(x) = x^n $ has no convergent subsequence in C[0, 1]. The additional property of equicontinuity is needed to characterize compact subsets of such spaces.
Topological and Metric Spaces
Published in J. Tinsley Oden, Leszek F. Demkowicz, Applied Functional Analysis, 2017
J. Tinsley Oden, Leszek F. Demkowicz
Ulisse Dini (1845–1918) (comp. Lemma 4.9.2) was an Italian mathematician. The notion of equicontinuity was introduced by an Italian mathematician, Giulio Ascoli (1843–1896) in 1884, and the Arzelà–Ascoli Theorem was proved in 1889 by another Italian, Cesare Arzelà (1847–1912). Andrei Nikolaevich Kolmogorov (1903–1987), a Russian mathematician, was the founder of the modern theory of probability. Among other fields, he is also famous for his contributions to topology and the theory of turbulence in fluid mechanics. Swedish mathematician Ivar Fredholm (1866–1927) was the founder of the modern theory of integral equations. His famous paper, published in 1903 in Acta Mathematica, started operator theory. Johannes Kepler (1571–1630) was a German mathematician, astronomer, and astrologer. Picard’s method is due to a French mathematician, Charles Émile Picard (1856–1941). The Contraction Map Theorem was proved by Polish mathematician Stefan Banach (1892–1945) (Chapter 5).
Equicontinuity and Li–Yorke pairs of dendrite maps
Published in Dynamical Systems, 2020
f is called equicontinuous (or stable in the sense of Lyapunov) at if for any there is such that for all provided that . The map f is equicontinuous if it is equicontinuous at any point of X. f is called uniformly equicontinuous if for any there is such that for all provided that . When the whole space is compact, the definitions of equicontinuity and uniform equicontinuity coincide. Notice that, for any positive integer i, if f is equicontinuous at x then f is so at . For any , the map f is equicontinuous at x if and only if is equicontinuous at x.