Ascoli-Arzelà theorem

Let $\\Omega$ be a bounded subset of $\\mathbb{R}^{n}$ and $(f_{k})$ a sequence of functions $f_{k}\\colon\\Omega\\to\\mathbb{R}^{m}$ . If $\\{f_{k}\\}$ is equibounded and uniformly equicontinuous then there exists a uniformly convergent subsequence $(f_{k_{j}})$ .

A more abstract (and more general) version is the following.

Let $X$ and $Y$ be totally bounded metric spaces and let $F\\subset\\mathcal{C}(X,Y)$ be an uniformly equicontinuous family of continuous mappings from $X$ to $Y$ .Then $F$ is totally bounded (with respect to the uniform convergence metric induced by $\\mathcal{C}(X,Y)$ ).

Notice that the first version is a consequence of the second.Recall, in fact, that a subset of a complete metric space is totally bounded if and only if its closure is compact (or sequentially compact).Hence $\\Omega$ is totally bounded and all the functions $f_{k}$ have image in a totally bounded set. Being $F=\\{f_{k}\\}$ totally bounded means that $\\overline{F}$ is sequentially compact and hence $(f_{k})$ has a convergent subsequence.

单词	AscoliArzelaTheorem
释义	Ascoli-Arzelà theorem Let $\\Omega$ be a bounded subset of $\\mathbb{R}^{n}$ and $(f_{k})$ a sequence of functions $f_{k}\\colon\\Omega\\to\\mathbb{R}^{m}$ . If $\\{f_{k}\\}$ is equibounded and uniformly equicontinuous then there exists a uniformly convergent subsequence $(f_{k_{j}})$ . A more abstract (and more general) version is the following. Let $X$ and $Y$ be totally bounded metric spaces and let $F\\subset\\mathcal{C}(X,Y)$ be an uniformly equicontinuous family of continuous mappings from $X$ to $Y$ .Then $F$ is totally bounded (with respect to the uniform convergence metric induced by $\\mathcal{C}(X,Y)$ ). Notice that the first version is a consequence of the second.Recall, in fact, that a subset of a complete metric space is totally bounded if and only if its closure is compact (or sequentially compact).Hence $\\Omega$ is totally bounded and all the functions $f_{k}$ have image in a totally bounded set. Being $F=\\{f_{k}\\}$ totally bounded means that $\\overline{F}$ is sequentially compact and hence $(f_{k})$ has a convergent subsequence.
随便看	LebesgueIntegralOverASubsetOfTheMeasureSpace Lebesgue integral over a subset of the measure space LebesgueMeasure Lebesgue measure LebesgueNumberLemma Lebesgue number lemma LebesgueOuterMeasure Lebesgue outer measure LectureNotesOnDeterminants lecture notes on determinants LectureNotesOnPolynomialInterpolation lecture notes on polynomial interpolation LectureNotesOnTheCayleyHamiltonTheorem lecture notes on the Cayley-Hamilton theorem LeechLattice Leech lattice LeftAndRightCosetsInADoubleCoset left and right cosets in a double coset LeftAndRightUnityOfRing left and right unity of ring LeftFunctionNotation left function notation LeftHandRule left hand rule LeftIdentityAndRightIdentity