Dini's theorem

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In the mathematical field of analysis, Dini's theorem says that if a monotone sequence of continuous functions converges pointwise on a compact space and if the limit function is also continuous, then the convergence is uniform.[1]

Formal statement[edit]

If is a compact topological space, and is a monotonically increasing sequence (meaning for all and ) of continuous real-valued functions on which converges pointwise to a continuous function , then the convergence is uniform. The same conclusion holds if is monotonically decreasing instead of increasing. The theorem is named after Ulisse Dini.[2]

This is one of the few situations in mathematics where pointwise convergence implies uniform convergence; the key is the greater control implied by the monotonicity. The limit function must be continuous, since a uniform limit of continuous functions is necessarily continuous.

Proof[edit]

Let be given. For each , let , and let be the set of those such that . Each is continuous, and so each is open (because each is the preimage of the open set under , a continuous function). Since is monotonically increasing, is monotonically decreasing, it follows that the sequence is ascending (i.e. for all ). Since converges pointwise to , it follows that the collection is an open cover of . By compactness, there is a finite subcover, and since are ascending the largest of these is a cover too. Thus we obtain that there is some positive integer such that . That is, if and is a point in , then , as desired.

Notes[edit]

  1. ^ Edwards 1994, p. 165. Friedman 2007, p. 199. Graves 2009, p. 121. Thomson, Bruckner & Bruckner 2008, p. 385.
  2. ^ According to Edwards 1994, p. 165, "[This theorem] is called Dini's theorem because Ulisse Dini (1845–1918) presented the original version of it in his book on the theory of functions of a real variable, published in Pisa in 1878.".

References[edit]

  • Bartle, Robert G. and Sherbert Donald R.(2000) "Introduction to Real Analysis, Third Edition" Wiley. p 238. – Presents a proof using gauges.
  • Edwards, Charles Henry (1994) [1973]. Advanced Calculus of Several Variables. Mineola, New York: Dover Publications. ISBN 978-0-486-68336-2.
  • Graves, Lawrence Murray (2009) [1946]. The theory of functions of real variables. Mineola, New York: Dover Publications. ISBN 978-0-486-47434-2.
  • Friedman, Avner (2007) [1971]. Advanced calculus. Mineola, New York: Dover Publications. ISBN 978-0-486-45795-6.
  • Jost, Jürgen (2005) Postmodern Analysis, Third Edition, Springer. See Theorem 12.1 on page 157 for the monotone increasing case.
  • Rudin, Walter R. (1976) Principles of Mathematical Analysis, Third Edition, McGraw–Hill. See Theorem 7.13 on page 150 for the monotone decreasing case.
  • Thomson, Brian S.; Bruckner, Judith B.; Bruckner, Andrew M. (2008) [2001]. Elementary Real Analysis. ClassicalRealAnalysis.com. ISBN 978-1-4348-4367-8.