Fredholm error bound for half-order product integration

Given: AC, the continuous $a$, the operators $K,K_n$, an exact solution $u$, and the uniform second-derivative bound in the Statement.

The Axiom of Choice and AC supplies the countable and dependent choices used in Banach integration: AC supplies the Countable Choice and Dependent Choice needed by the local Arzelà–Ascoli result.

Piecewise-linear product integration for a half-order diagonal singularity: $K$ and $K_n$ are the improper-integral and weighted-hat interpolation operators, and $K_n$ has the finite-node form ${\sum }_j{\psi }_j(x)u(y_j)$ with coefficient functions ${\psi }_j(x)=w_{n,j}(x)a(x,y_j)$.

Stability, collective compactness, and consistency of half-order product integration: $K$ and each $K_n$ are bounded linear maps, $K$ is compact, $K_n$ is collectively compact and pointwise norm convergent to $K$, each coefficient function ${\psi }_j$ is continuous, and $\mathrm{\parallel }{K}_{n}u-Ku{\mathrm{\parallel }}_{\mathrm{\infty }}\le \sqrt{2}{M}_2{h}^2\mathrm{/}4$.

Collective compactness gives eventual Fredholm invertibility and solution error bounds: under AC and injectivity of $I-K$, the operators $I-K_n$ are boundedly invertible for all sufficiently large $n$ with one uniform inverse bound, and their solution errors satisfy the residual estimate.

Finite-node Nyström equations are equivalent to their reconstructed equations: bijective solvability of the reconstructed equation is equivalent to nonsingularity of its finite grid matrix.

On the nonempty compact interval $[0,1]$ (Heine-Borel in ${\mathbb{R}}^n$: with the Euclidean metric a subset of ${\mathbb{R}}^n$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line), let $X=C([0,1],\mathbb{R})$ with its supremum norm. Pointwise addition and real scalar multiplication preserve continuity. The supremum is finite by $C(K,\mathbb{R})$ is complete in the supremum metric for every nonempty compact metric space $K$; the pointwise absolute-value inequalities give $\mathrm{\parallel }u+v{\mathrm{\parallel }}_{\mathrm{\infty }}\le \mathrm{\parallel }u{\mathrm{\parallel }}_{\mathrm{\infty }}+\mathrm{\parallel }v{\mathrm{\parallel }}_{\mathrm{\infty }}$ and $\mathrm{\parallel }cu{\mathrm{\parallel }}_{\mathrm{\infty }}=\mathrm{\mid }c\mathrm{\mid }\mathrm{\parallel }u{\mathrm{\parallel }}_{\mathrm{\infty }}$, and $\mathrm{\parallel }u{\mathrm{\parallel }}_{\mathrm{\infty }}=0$ holds exactly when $u=0$. Thus this is a real norm (A norm on a real vector space, the induced metric, and the dictionary with the metric axioms). Its induced metric is the complete supremum metric of $C(K,\mathbb{R})$ is complete in the supremum metric for every nonempty compact metric space $K$, so $X$ is Banach (Banach space). The constant function $1$ is a nonzero member of $X$ because the interval is nonempty (The space $C(K,\mathbb{R})$ of continuous real-valued functions on a nonempty compact metric space).