We prove that if X is an infinite Tychonoff space, then no compactification of Cp(X) or Ck(X) has a countably compact remainder. This motivates the question on whether the result continues to hold for all locally convex topological vector spaces over the reals. We will show that this is not the case for many of such spaces.
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Банах et al. (2026) studied this question.
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