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Keywords:
compact Hausdorff space $X$; the sup-norm algebra $C(X)$ of all complex-valued continuous functions on $X$; its closed subalgebras (called function algebras); its closed subspaces (called function spaces); measure orthogonal to a function algebra or to a function space
compact Hausdorff space $X$; the sup-norm algebra $C(X)$ of all complex-valued continuous functions on $X$; its closed subalgebras (called function algebras); its closed subspaces (called function spaces); measure orthogonal to a function algebra or to a function space
Summary:
It is well known that any function algebra has an essential set. In this note we define an essential set for an arbitrary function space (not necessarily algebra) and prove that any function space has an essential set.
References:
[1] Bear H.S.: Complex function algebras. Trans. Amer. Math. Soc. 90 (1959), 383-393. MR 0107164 | Zbl 0086.31602
[2] Hoffman K., Singer I.M.: Maximal algebras of continuous functions. Acta Math. 103 (1960), 217-241. MR 0117540 | Zbl 0195.13903
[3] Glicksberg I.: Measures orthogonal to algebras and sets of antisymmetry. Trans. Amer. Math. Soc. 98 (1962), 415-435. MR 0173957 | Zbl 0111.11801
[4] Čerych J.: On essential sets of function algebras in terms of their orthogonal measures. Comment. Math. Univ. Carolinae 36.3 (1995), 471-474. MR 1364487
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