Positive right and adjoint eigenfunctions have the same eigenvalue

Use the adjoint identity with $f=\chi$ and $g=\varphi$. Substituting the eigenvalue equations and using linearity in F7 gives $(\lambda -{\lambda }^{†}){\int }_S\chi \varphi d{\mu }_h=0$. Since $S$ is nonempty, $\chi \varphi d{\mu }_h$ is a nonzero positive smooth density. Its support is all of compact $S$, so F7 applies, with countable choice supplied by F8, and gives ${\int }_S\chi \varphi d{\mu }_h>0$. Hence the eigenvalues agree.

By F3 the smooth real functions $\varphi$ and $\chi$ are continuous. Apply F2 to them on the nonempty compact manifold $S$. Each attains its minimum at a point where its value is positive, giving ${\min }_S\varphi >0$ and ${\min }_S\chi >0$. The map $r:(0,\mathrm{\infty })\to \mathbb{R}$, $r(s)=1\mathrm{/}s$, is smooth: its derivative of order $j\ge 0$ is $(-1{)}^{j}j!s^{-j-1}$. Since $\varphi$ and $\chi$ take values in this open interval, F4 makes $1\mathrm{/}\varphi =r\circ \varphi$ and $1\mathrm{/}\chi =r\circ \chi$ smooth. They satisfy $1\mathrm{/}\varphi \le ({\min }_S\varphi {)}^{-1}$ and $1\mathrm{/}\chi \le ({\min }_S\chi {)}^{-1}$. For $u\in \{1\mathrm{/}\varphi ,1\mathrm{/}\chi \}$ define ${\mathrm{\nabla }}^{0}u=u$ and ${\mathrm{\nabla }}^{j+1}u=\mathrm{\nabla }({\mathrm{\nabla }}^{j}u)$ using F5 and F6. Function-linearity in the differentiating vector field makes ${\mathrm{\nabla }}^{j+1}u$ a covariant $(j+1)$-tensor; the smooth dual and product connections show inductively that all these tensors have smooth local components. For every fixed derivative order, the squared Riemannian norm of the corresponding covariant derivative of either smooth reciprocal is a smooth real function: in coordinates it is a finite contraction of smooth tensor components with the smooth inverse metric. It is continuous by F3 and bounded by F2, so its nonnegative square root, the Riemannian norm itself, is continuous and bounded as well. This proves all assertions.