Английская Википедия: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]
Formal statement
If <math>X</math> is a compact topological space, and <math>(f_n)_{n\in\mathbb{N}}</math> is a monotonically increasing sequence (meaning <math>f_n(x)\leq f_{n+1}(x)</math> for all <math>n\in\mathbb{N}</math> and <math>x\in X</math>) of continuous real-valued functions on <math>X</math> which converges pointwise to a continuous function <math>f\colon X\to \mathbb{R}</math>, then the convergence is uniform. The same conclusion holds if <math>(f_n)_{n\in\mathbb{N}}</math> 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. The continuity of the limit function cannot be inferred from the other hypothesis (consider <math>x^n</math> in <math>[0,1]</math>.)
Proof
Let <math>\varepsilon > 0</math> be given. For each <math>n\in\mathbb{N}</math>, let <math>g_n=f-f_n</math>, and let <math>E_n</math> be the set of those <math>x\in X</math> such that <math>g_n(x)<\varepsilon</math>. Each <math>g_n</math> is continuous, and so each <math>E_n</math> is open (because each <math>E_n</math> is the preimage of the open set <math>(-\infty, \varepsilon)</math> under <math>g_n</math>, a continuous function). Since <math>(f_n)_{n\in\mathbb{N}}</math> is monotonically increasing, <math>(g_n)_{n\in\mathbb{N}}</math> is monotonically decreasing, it follows that the sequence <math>E_n</math> is ascending (i.e. <math>E_n\subset E_{n+1}</math> for all <math>n\in\mathbb{N}</math>). Since <math>(f_n)_{n\in\mathbb{N}}</math> converges pointwise to <math>f</math>, it follows that the collection <math>(E_n)_{n\in\mathbb{N}}</math> is an open cover of <math>X</math>. By compactness, there is a finite subcover, and since <math>E_n</math> are ascending the largest of these is a cover too. Thus we obtain that there is some positive integer <math>N</math> such that <math>E_N=X</math>. That is, if <math>n>N</math> and <math>x</math> is a point in <math>X</math>, then <math>|f(x)-f_n(x)|<\varepsilon</math>, as desired.
Notes
References
- Bartle, Robert G. and Sherbert Donald R.(2000) "Introduction to Real Analysis, Third Edition" Wiley. p 238. – Presents a proof using gauges.
- Шаблон:Cite book
- Шаблон:Cite book
- Шаблон:Cite book
- 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.
- Шаблон:Cite book
- ↑ Шаблон:Harvnb. Шаблон:Harvnb. Шаблон:Harvnb. Шаблон:Harvnb.
- ↑ According to Шаблон:Harvnb, "[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".
