= Finite-dimensional embedding of a complex vector bundle
{title2=$J(v)=(\sqrt{\rho_i(x)}\,t_i(v))_i$}
For finitely many orthonormal <local trivializations> $t_i$ and a subordinate <partition of unity> $\rho_i$, the displayed map embeds a rank-k <complex vector bundle> isometrically in $X\times\mathbb C^{kr}$. Outside the support of each $\rho_i$ its component is zero. Its continuous image planes give a finite-dimensional <Grassmannian> map and identify the original bundle with a tautological pullback.
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