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In this paper we study the family of embeddings $\Phi_t$ of a compact$RCD^*(K,N)$ space $(X,d,m)$ into $L^2(X,m)$ via eigenmaps. Extending part ofthe classical results by B\'erard, B\'erard-Besson-Gallot, known for closedRiemannian manifolds, we prove convergence as $t\downarrow 0$ of the rescaledpull-back metrics $\Phi_t^*g_{L^2}$ in $L^2(X,m)$ induced by $\Phi_t$. Moreoverwe discuss the behavior of $\Phi_t^*g_{L^2}$ with respect to measuredGromov-Hausdorff convergence and $t$. Applications include the quantitative$L^p$-convergence in the noncollapsed setting for all $p<\infty$, a result neweven for closed Riemannian manifolds and Alexandrov spaces.

Embedding of $RCD^*(K,N)$ spaces in $L^2$ via eigenfunctions