In mathematics, Grauert's approximation theorem, due to Grauert, is an analog of Whitney’s approximation theorem for real-analytic maps. It states: with respect to the Whitney topology (also known as strong topology), the space of real-analytic maps between real-analytic manifolds is dense in the space of smooth maps between those manifolds. In the compact case, the theorem is due to Morrey. The case when there is an analytic Riemannian metric is due to Bochner.
See also Mapping space#Smooth mappings
References
Grauert, Hans (1958). "On Levi's Problem and the Imbedding of Real-Analytic Manifolds". Annals of Mathematics. 68 (2): 460–472. doi:10.2307/1970257. ISSN 0003-486X. Hirsch, Morris W. (1976). Differential topology. New York Heidelberg Berlin: Springer-Verlag. ISBN 978-1-4684-9449-5.