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Übersetzung für 'Arzelà-Ascoli' von Englisch nach Deutsch

Arzelà-Ascoli theorem

Satz {m} von Arzelà-Ascolimath.

• Satz {m} von Arzelà-Ascoli = Arzelà-Ascoli theorem

• In 1889, Italian mathematician Cesare Arzelà generalized Ascoli's Theorem into the Arzelà–Ascoli theorem, a practical sequential compactness criterion of functions.
• These include theorems about compactness of certain spaces such as the Banach–Alaoglu theorem on the weak-* compactness of the unit ball of the dual space of a normed vector space, and the Arzelà–Ascoli theorem characterizing the sequences of functions in which every subsequence has a uniformly convergent subsequence.
• The Arzelà–Ascoli theorem and the Peano existence theorem exemplify applications of this notion of compactness to classical analysis.
• Cesare Arzelà (6 March 1847 – 15 March 1912) was an Italian mathematician who taught at the University of Bologna and is recognized for his contributions in the theory of functions, particularly for his characterization of sequences of continuous functions, generalizing the one given earlier by Giulio Ascoli in the Arzelà–Ascoli theorem.
• An example is the Arzelà–Ascoli theorem. Other cases of interest relate to uniform integrability, and the concept of normal family in complex analysis.

• Then Arzelà–Ascoli theorem states that a subset of "C"("X") is compact if and only if it is closed, uniformly bounded and equicontinuous.
• So by the Arzelà–Ascoli theorem, passing to a subsequence if necessary, it can be assumed that both "f'n" and "g'n" converge uniformly on compacta to "f" and "g".
• The essence of the proof is that once the first half is established, one may directly apply the Arzelà–Ascoli theorem, in the context of the compact metric space [...] to a sequence of piecewise continuously-differentiable unit-speed curves from [...] to [...] whose lengths approximate [...] The resulting subsequential limit is the desired geodesic.