Pervasive algebras of analytic functions

Let X be a compact Hausdorff topological space and C(X, C) (respectively, C(X, R)) the Banach algebra of all continuous complex-valued (respectively, real-valued) functions on X endowed with the uniform norm. A function space S on X is a closed subspace of C(X, C). We denote by closC(E, C) S the closure in C(E, C) of the function space S, where E is a closed subset of X. Similarly, we denote by closC(E, R) S the closure in C(E, R) of the real subspace S of C(X, R)