Neural-operator quadratures in the uniform norm and compatible Banach norms

Write $I(v)={\int }_D\kappa (\cdot ,y)v(y)dy$ for one compatible Banach layer, and let $C$ be a compact subset of its input space $X$. Its image in $C(\bar{D})$ is compact: pulling an open cover back through the continuous inclusion and using compactness proves this directly. Thus $R={\sup }_{v\in C}\mathrm{\parallel }v{\mathrm{\parallel }}_{\mathrm{\infty }}<\mathrm{\infty }$. This family has a common modulus of continuity tending to zero. Indeed a finite $\eta$-net in the supremum norm and uniform continuity of its finitely many functions give $\mathrm{\mid }v(y)-v(z)\mathrm{\mid }\le 2\eta +{\max }_j\mathrm{\mid }{v}_j(y)-v_j(z)\mathrm{\mid }$ for every $v\in C$. Uniform continuity of the finitely many functions follows by the compact-domain subsequence contradiction. Let $B={\sup }_y\mathrm{\parallel }\kappa (\cdot ,y){\mathrm{\parallel }}_{\mathcal{L}({\mathbb{R}}^q,Y)}$ and let ${\omega }_{\kappa }$ be its norm modulus. Compactness similarly proves $B<\mathrm{\infty }$ and ${\omega }_{\kappa }(h)\to 0$. Therefore the $Y$-valued integrands ${\mathrm{\Phi }}_v(y)=\kappa (\cdot ,y)v(y)$ have common modulus bounded by $R{\omega }_{\kappa }(h)+B{\omega }_C(h)$.

A continuous $Y$-valued function $\mathrm{\Phi }$ on $\bar{D}$ has an integral in $Y$: volume-weighted step sums on partitions of diameter $h$ form a Cauchy family, since comparing two sums on their common refinement bounds their difference by $\mathrm{\mid }D\mathrm{\mid }{\omega }_{\mathrm{\Phi }}(h+h^{\mathrm{\prime }})$. Completeness of $Y$ gives a limit, independent of tags and partitions. Each coordinate of point evaluation on $Y$ is bounded by its specified continuous inclusion into $C({\bar{D}}^{\mathrm{\prime }};{\mathbb{R}}^{q_{t+1}})$; explicitly, $\mathrm{\mid }{u}_j(x)\mathrm{\mid }\le c_Y\mathrm{\parallel }u{\mathrm{\parallel }}_Y$. Applying these bounded functionals to the convergent sums therefore identifies the limit with the pointwise integral in every coordinate. Comparing a fixed tagged sum to refinements and taking the limit gives $\mathrm{\parallel }{Q}_n(v)-I(v){\mathrm{\parallel }}_Y\le \mathrm{\mid }D\mathrm{\mid }[R{\omega }_{\kappa }(h_n)+B{\omega }_C(h_n)]$ uniformly for $v\in C$. Also $\mathrm{\parallel }{Q}_n(v)-Q_n(w){\mathrm{\parallel }}_Y\le \mathrm{\mid }D\mathrm{\mid }B\mathrm{\parallel }v-w{\mathrm{\parallel }}_{\mathrm{\infty }}\le \mathrm{\mid }D\mathrm{\mid }B{c}_X\mathrm{\parallel }v-w{\mathrm{\parallel }}_X$, where $c_X$ is the embedding bound. These estimates use only measurable disjoint cells, their volumes summing to $\mathrm{\mid }D\mathrm{\mid }$, and their diameter bound.

Suppose inductively that the discrete inputs $v_n(a)$ converge in $X$ to the continuum inputs $v(a)$ uniformly for $a$ in the original compact set $K$. The continuum map is continuous by the bound for $I$ from step 2.1 and continuity of all other maps, so $C=v(K)$ is compact. Comparing preactivations at $v_n(a)$ and $v(a)$, the quadrature Lipschitz bound and the bounded local-map norm give an upper bound $(\mathrm{\mid }D\mathrm{\mid }B{c}_X+\mathrm{\parallel }W\mathrm{\parallel })\mathrm{\parallel }{v}_n(a)-v(a){\mathrm{\parallel }}_X+\mathrm{\parallel }{Q}_n(v(a))-I(v(a)){\mathrm{\parallel }}_Y$, which tends to zero uniformly in $a$. The continuum preactivations form a compact set $P\subset Y$. A continuous map $T:Y\to X^{\mathrm{\prime }}$ takes perturbations tending uniformly to zero around $P$ to uniformly vanishing output errors: for a prescribed $\epsilon$, cover $P$ by finitely many balls $B(p_i,r_i)$ on each of which $\mathrm{\parallel }T(z)-T(p_i)\mathrm{\parallel }<\epsilon \mathrm{/}2$ for $\mathrm{\parallel }z-p_i\mathrm{\parallel }<3r_i$, and use a positive perturbation bound smaller than ${\min }_i{r}_i$. If $p\in B(p_i,r_i)$ and $\mathrm{\parallel }z-p\mathrm{\parallel }<{\min }_i{r}_i$, both images are within $\epsilon \mathrm{/}2$ of $T(p_i)$. Apply this to the activation. The same argument treats continuous lifting or projection maps. Starting with the identical input and proceeding through the finitely many stages proves uniform convergence in the final Banach norm.

For the supremum-norm architecture take every relevant $X$ and $Y$ to be the continuous-function space with its supremum norm. Joint continuity of $\kappa (x,y)$ on the compact product implies continuity of $y\mapsto \kappa (\cdot ,y)$ in that norm, by its uniform modulus on the product. Finite matrix local terms are bounded in that norm. For any fixed continuous scalar activation, its induced map on continuous functions is continuous in the supremum norm: near a function of norm $R$, all scalar arguments lie in $[-R-1,R+1]$, where the activation is uniformly continuous. Pointwise finite networks are finite compositions of these continuous maps. Thus all conditions in steps 1.1–2.1 hold, proving the first clause and then its continuously embedded input-space version.

Each raw map is continuous between its specified spaces: finite quadrature sums use bounded point evaluations and bounded kernel maps, and all local, activation, lifting and projection maps are continuous by hypothesis. For continuous inputs, apply the explicit reconstructions in [F3]; steps 1.1–4.1 supply the raw compact-uniform convergence required there. The bridge gives finite-dimensional sampling and output reconstruction with the same continuum parameters and output supremum convergence. In the general Banach case use the supplied reconstructions converging in its actual input norm and apply the first clause of the same bridge; its compact-union argument preserves exactly the output norm proved in step 3.1. For an integral-only first layer after a general lifting $P$, define $S_n(a)=(P(a)({\xi }_{n,j}){)}_j$. For an arbitrary vector $z$ in this finite-dimensional space, form the first field ${\sum }_j{\omega }_{n,j}\kappa (\cdot ,{\xi }_{n,j})z_j+b$ and apply all subsequent stages to define $R_n(z)$. Compatibility makes this a continuous output-valued map, and the raw map equals $R_n{S}_n$ exactly. This is the construction of [F3], with the lifted field as input. A pointwise lift computes $S_n$ from the original input samples; a nonlocal lift need not do so. The explicitly specified sampling map and identity finite map give the asserted discretization family in either case.