Smooth-solution error bound for composite midpoint Nyström discretization

Given: AC, $k$, $K$, $K_N$, $u$, $f$, $f_N$, and $M_2$ as in the Statement.

Nyström quadrature operators and their reconstructed equations: the Nyström equation is $(I-K_N)u_N=f_N$.

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

A continuous kernel defines a compact integral operator on C of an interval: the continuous-kernel operator $K$ is compact; AC supplies its Countable Choice and Dependent Choice hypotheses by AC supplies the countable and dependent choices used in Banach integration.

Composite midpoint Nyström operators are collectively compact and consistent: each composite midpoint operator is bounded linear, the family is collectively compact, and $K_{N}x\to Kx$ in norm for every $x\in X$.

Collective compactness gives eventual Fredholm invertibility and solution error bounds: under AC, compactness of $K$, collective compactness and pointwise convergence of $K_N$, and injectivity of $I-K$ give eventual bounded inverses with a uniform bound and the residual solution estimate.

Composite midpoint error bound for a twice continuously differentiable function: for a $C^2$ real integrand $g$ on $[0,1]$, the composite midpoint rule satisfies $\mathrm{\mid }{\int }_0^{1}g-Q_{N}g\mathrm{\mid }\le h^2\mathrm{\parallel }{g}^{\mathrm{\prime }\mathrm{\prime }}{\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{/}24$.

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 $1$ belongs to $X$ and is nonzero by the definition of continuous real-valued functions (The space $C(K,\mathbb{R})$ of continuous real-valued functions on a nonempty compact metric space), so $X\mathrm{\ne }\{0\}$.