  An $L^2$ version of the classical Denjoy-Carleman theorem regarding
quasi-analytic functions was proved by P. Chernoff on $\mathbb R^n$ using
iterates of the Laplacian. We give a simple proof of this theorem which
generalizes the result on $\mathbb R^n$ for any $p\in [1, 2]$. We then extend
this result to Riemannian symmetric spaces of compact and noncompact type for
$K$-biinvariant functions.
