Dini's theorem
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]}
Contents
Formal statement
If X is a compact topological space, and { f_{n} } is a monotonically increasing sequence (meaning f_{n}(x) ≤ f_{n+1}(x) for all n and x) of continuous real-valued functions on X which converges pointwise to a continuous function f, then the convergence is uniform. The same conclusion holds if { f_{n} } 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. Note also that the limit function must be continuous, since a uniform limit of continuous functions is necessarily continuous.
Proof
Let ε > 0 be given. For each n, let g_{n} = f − f_{n}, and let E_{n} be the set of those x ∈ X such that g_{n}( x ) < ε. Each g_{n} is continuous, and so each E_{n} is open (because each E_{n} is the preimage of an open set under g_{n}, a nonnegative continuous function). Since { f_{n} } is monotonically increasing, { g_{n} } is monotonically decreasing, it follows that the sequence E_{n} is ascending. Since f_{n} converges pointwise to f, it follows that the collection { E_{n} } is an open cover of X. By compactness, there is a finite subcover, and since E_{n} are ascending the largest of these is a cover too. Thus we obtain that there is some positive integer N such that E_{N} = X. That is, if n > N and x is a point in X, then |f( x ) − f_{n}( x )| < ε, as desired.
Notes
- ^ Edwards 1994, p. 165. Friedman 2007, p. 199. Graves 2009, p. 121. Thomson, Bruckner & Bruckner 2008, p. 385.
- ^ 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
- 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.