Real and finite-dimensional Euclidean Ascoli–Arzelà criteria

By [L3], both $\mathbb{R}$ and ${\mathbb{R}}^n$ are proper metric spaces: a closed bounded subset is compact.

By [L9] and [L10], $\bar{\rho }(f,g)=\min \{d_{\mathrm{\infty }}(f,g),1\}$, so balls of radius below $1$ agree and the supremum-metric topology is the uniform topology. In the real case, compactness then implies uniform closedness by [L4], and [L5] gives equicontinuity and pointwise boundedness. Conversely, uniform closedness plus those two conditions makes the compact closure supplied by [L5] equal to the family. The subsequence conclusion is [L6], and the stated uniform equicontinuity and common value bound are exactly [L7] and [L8].

For the Euclidean common bound, fix an equicontinuous, pointwise bounded family $\mathcal{F}\subseteq C(X,{\mathbb{R}}^n)$ and take the tolerance to be $1$. Equicontinuity of $\mathcal{F}$ gives, at every $x\in X$, a neighbourhood $U_x$ on which $\parallel f(y)-f(x){\parallel }_2<1$ for all $f\in \mathcal{F}$; Choice, which the Given supplies, licenses the family $(U_x{)}_{x\in X}$. Compactness of $X$ gives a finite subcover $U_{x_1},\dots ,U_{x_m}$.

Apply [L1] and [L2] to the proper target ${\mathbb{R}}^n$. This proves both numbered vector-valued assertions, including both directions of assertion 1.

Pointwise boundedness of the family $\mathcal{F}$ fixed in step 1.3 gives, for each of the finitely many $i\le m$, a number $M_i$ with $\parallel f(x_i){\parallel }_2\le M_i$ for every $f\in \mathcal{F}$. For $y\in X$ take $i$ with $y\in U_{x_i}$; then $\parallel f(y){\parallel }_2\le \parallel f(y)-f(x_i){\parallel }_2+\parallel f(x_i){\parallel }_2<1+M_i\le 1+{\max }_i{M}_i$. Thus one Euclidean bound serves all $f\in \mathcal{F}$ and all $y\in X$, and the converse from a common bound to pointwise boundedness is immediate.