Theorem

Dini's Theorem

Analysis

Dini's Theorem states that if a sequence of continuous functions on a compact space converges pointwise and monotonically to a continuous limit function, then the convergence is in fact uniform. Named for Ulisse Dini, it is a standard result of real analysis identifying one of the more common conditions under which pointwise convergence upgrades to the stronger, more useful property of uniform convergence.

Facts

Statement

If a monotone sequence of continuous functions converges pointwise on a compact space and the limit function is also continuous, then the convergence is uniform. 1

Proof Year

1878 1

Classification

Statement Form

Characterization Theorem 1

Connections

Has Statement Form

Characterization

In Branch

Analysis

Source Dini's Theorem (Wikipedia)

Sources

1. Dini's Theorem (Wikipedia)

Wikimedia Foundation

Lead section, theorem statement sentence
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.
References
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.
In Branch: Analysis, Lead paragraph
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.

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Dini's Theorem

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