Lagmental Vicfred

Arzelà--Ascoli Converts Equicontinuity into Compactness by Vicfred

tags: mathematics, uniform convergence, compactness, approximation

On a compact space, uniform boundedness and equicontinuity give a uniformly convergent subsequence. I want the notation, the mechanism, and the failure mode visible at the same time.

Objects and notation

A sequence \(f_n:X\to\mathbf R\) converges uniformly to \(f\) when \(\sup_{x\in X}|f_n(x)-f(x)|\to0\). The supremum norm captures one error bound valid everywhere.

$$ \mathcal F\subset C(K),\qquad K\ \text{compact} $$

The typography mirrors the proof: first declare \(\mathsf D\), then state \(\mathsf C\). The symbol \(\Longrightarrow\) below is a logical dependency, not extra mathematical structure.

$$ \mathcal F\ \text{uniformly bounded and equicontinuous}\Longrightarrow\overline{\mathcal F}\ \text{compact in }\|\cdot\|_\infty $$

Push the symbols

Now evaluate one representative case. The result should agree with the structural law above, but it is obtained without assuming the conclusion.

$$ \forall\varepsilon>0\ \exists\delta>0:\ d(x,y)<\delta\Longrightarrow\sup_{f\in\mathcal F}|f(x)-f(y)|<\varepsilon $$

Structural reading

The invariant statement is the one that does not depend on a convenient choice of coordinates, representatives, basis, or enumeration.

$$ \begin{aligned} \mathsf{D}\;&:\quad \mathcal F\subset C(K),\qquad K\ \text{compact},\\[5pt] \mathsf{C}\;&:\quad \mathcal F\ \text{uniformly bounded and equicontinuous}\Longrightarrow\overline{\mathcal F}\ \text{compact in }\|\cdot\|_\infty. \end{aligned} $$

A hypothesis worth keeping

Pointwise convergence allows the bad point to move with \(n\). Continuity, integration, and differentiation survive limits under different uniform hypotheses.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \mathcal F\ \text{uniformly bounded and equicontinuous}\Longrightarrow\overline{\mathcal F}\ \text{compact in }\|\cdot\|_\infty \end{gathered}} $$

The notation is dense, but it is doing honest work: every delimiter records scope and every index records dependence. Removing one should require a mathematical reason.

This article was posted on Thu 15 May 2014. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.