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Function spaces have essential sets. (English) Zbl 0937.46048
Let $$X$$ be a compact Hausdorff topological space and let $$C(X)$$ be the algebra of continuous complex-valued functions on $$X$$. A closed subalgebra of $$C(X)$$ which separates points of $$X$$ and which contains constant functions is called a function algebra. A closed subspace of $$C(X)$$ which separates points of $$X$$ is called a function space. It is known that any function algebra has an essential set. In the paper an essential set is constructed for every function space which does not necessarily have to be an algebra.
##### MSC:
 46J10 Banach algebras of continuous functions, function algebras 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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