Let C(K) be the Banach space of real-valued continuous functions on a compact Hausdorff space with the supremum norm and let Xbea closed subspace of C(K) which separates points of K. Necessary and sufficient conditions are given for X to be the range of a projection of norm one in C(K). It is shown that the form of a projection of norm one is determined by a real-valued continuous function which is defined on a subset of K and which satisfies conditions imposed by X. When there is a projection

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... f norm one onto X, it is shown that there is a one-to-one correspondence between the continuous functions which satisfy the conditions imposed by X and the projections of norm one onto X. Using well-known results of Nachbin, Goodner and Kelley (see [2, p. 2]), it is easily demonstrated that if Sis an extremely disconnected compact Hausdorff space, then X, a closed subspace of C(S), is the range of a norm one projection in C(S) iff X itself is isometric to the continuous functions on an extremely disconnected compact Hausdorff space. The main purpose of this research is to give necessary and sufficient conditions for a closed separating subspace X of C(K), K a compact Hausdorff space, to be the range of a norm one projection in C(K). Theorem 3.1 supplies these conditions. Proposition 1.2 shows that the form of each contractive projection onto X can be given in terms of a real-valued continuous function, determined by the projection, which is defined on a subset of K. This continuous function satisfies certain conditions which depend only on the subspace X. Theorem 3.1 also shows that when there is a contractive projection onto X there is a one-to-one correspondence between contractive projections onto X and the continuous functions which satisfy these conditions imposed by X. We give another proof to the fact, proved in [3] , that the Banach space Y is isometric to the range of a contractive