S. J. Bernau has introduced the notion of an exchange subspace of an L p -space and has shown that the range of a contractive linear projection on an L p -space (1 ⩽ p < ∞, p ≠ 2) is an exchange subspace. In the present paper we define this notion for real Banach lattices with order continuous norm and prove among other things that fixed spaces of special regular operators on these spaces are exchange subspaces. As application we give a Korovkin theorem for sequences of contractions on real Banach lattices with an uniformly monotone norm.