This paper calculates \(KO^*({\mathbb C}P^n)\) and \(KO^*({\mathbb C}P^n \times {\mathbb C}P^m)\) as algebras over \(\pi_* KO\). This completes results of M. Fujii [Osaka J. Math. 4, 141-149 (1967; Zbl 0153.25201)] who computed the abelian group structure and the ring structure of the subalgebra of elements of even degree of \(KO^*({\mathbb C}P^n)\). The descriptions the author obtains, which are too lengthy to be give heren, are obtained using the Atiyah-Hirzebruch spectral sequence. The author points out an interesting consequence of these calculations: that the image of the map \(KO^*({\mathbb C}P^{n+m}) \to KO^*({\mathbb C}P^n \times {\mathbb C}P^m)\) induced by the map classifying the tensor product of the canonical line bundles is not contained in the image of the cross product and so that there is no formal group structure on \(KO^*({\mathbb C}P^\infty)\) analogous to that on \(K^*({\mathbb C}P^\infty)\).