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JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. Annals of Mathematics is collaborating with JSTOR to digitize, preserve and extend access to Annals of Mathematics. Introduction.If X is any topological space, let B(X) be the space of real bounded continuous functions on X, made into a Banach space by the usual norm II b I = supZExIb(x) 1.According to the Banach-Stone Theorem (see [2], [7], [3],),' if X is compact2 B(X) determines the topology of X, in the sense that if B(X1) is equivalent to B(X2) for compact Xi and X2 then Xi is homeomorphic to X2. In this paper we find that a certain type of closed linear subspace of B(X) is sufficient to determine the topology of a compact X; we give conditions under which there exists an X such that a given Banach space B is equivalent to such a subspace of B (X), and also conditions under which there exists a compact X such that B is equivalent to the whole of B(X). More specifically, let us define a subset D of B(X) to be comprpetely regu!ar (over X) if for every closed set K C X and every point XO E X -K, there exists a beD such that b(xo) = Il b 11, SUpxEK I b(x) I < 1l b 11 Then we shall find I. A proof that a completely regular closed linear subspace of B(X) for a compact X determines the topology of X, in the sense that if such a subspace of B(X1) is equivalent to such a subspace of B(X2) for compact Xi, X2, then XI is homeomorphic to X2 (see Theorem 4.2).II. Sufficient conditions that there exist an X such that a given Banach space B is equivalent to a closed completely regular linear subspace of B(X) (see Theorem 3.2); also necessary conditions for the existence of such a compact X (see Theorem 4.1).III. Necessary and sufficient conditions that a given Banach space B be equivalent to B(X) for some compact (or completely regular) X (see Theorem 5.1 and 5.2).Result I generalizes the Banach-Stone Theorem. In connection with II, Alaoglu [1] has pointed out that every Banach space B is equivalent to a closed linear subspace of B(X), where X is the unit sphere in the space B* conjugate to B, topologized by the weak topology. Of course in general this subspace is not completely regular over X. Result III is a characterization of the space of continuous functions.3 1 Numbers in brackets refer to the bibliography. 2 Throughout this paper a compact space means a Hausdorff space with the property that every covering by open sets contains a finite-sub-covering.