Numerable vector bundles admit bundle metrics

First suppose the numeration in [F1] is supplied. Transport the standard Euclidean or Hermitian form to $E{\mathrm{\mid }}_{U_i}$ and call it $h_i$. Define on each fiber $h_x(v,w)={\sum }_i{\rho }_i(x)h_{i,x}(v,w)$, taking the $i$th term to be zero off $U_i$. Since $\mathrm{supp}{\rho }_i\subseteq U_i$, this zero extension is continuous near every point outside $U_i$, and local finiteness makes the sum continuous.

Each summand is positive semidefinite. At every $x$, some ${\rho }_i(x)>0$ because the coefficients sum to one; for $v\mathrm{\ne }0$, the corresponding $h_{i,x}(v,v)>0$. Hence $h_x(v,v)>0$. The formula is symmetric bilinear over $\mathbb{R}$ or conjugate-symmetric sesquilinear over $\mathbb{C}$, so it is the required metric.

If $X$ is paracompact Hausdorff, apply [A1] to obtain DC and then [F2] to the linear chart cover of $E$. This supplies a numeration, so steps 1.1–2.1 give a metric. AC is used only through this invocation of the published partition theorem; with supplied data those two steps make no choices.