Teorema de Arzelà-Ascoli

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Em matemática, o teorema de Ascolí-Arzela é um importante resultado, com aplicações na análise real, análise funcional e em áreas afins tais como a teoria das equações diferenciais. Provém dos matemáticos italianos Cesare Arzelà e Giulio Ascoli.

Enunciado da versão real

Seja ${\mathfrak {F}}\,$ uma sequência de funções $f:[a,b]\to \mathbf {R} \,$ com as seguintes propriedades:

Equicontinuidade, ou seja, para cada $\varepsilon >0\,$ e cada $x\,$ no domínio, existe um $\delta >0\,$ tal que $|x-y|<\delta \Longrightarrow |f(x)-f(y)|<\varepsilon ,\forall f\in {\mathfrak {F}}\,$
Equilimitação, ou seja, existe uma constante $C\,$ tal que $|f(x)|<C,\forall x\in [a,b],\forall f\in {\mathfrak {F}}\,$

Então existe uma subseqüência $f_{n}(x)\,$ e uma função contínua $f(x)\,$ tal que $f_{n}(x)\,$ converge uniformemente para $f(x)\,$ .

De uma forma mais simples, o teorema pode ser enunciado da seguinte forma:

Considere uma sequência de funções contínuas $(f_{n}(x))\,$ definidas em um intervalo fechado [a,b] dos reais. Se essa sequência é uniformemente limitada e equicontínua, então existe uma subsequência que converge uniformemente.

Isso significa, por exemplo, que o teorema funciona para funções deriváveis tais que ela e sua derivada são uniformemente limitadas. Se a derivada segunda também é uniformemente limitada, as derivadas também convergem uniformemente.

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