Abstract

Cp(Y∣X)C_{p}\left( Y|X\right) denotes the real-valued continuous functions on Y⊆XY\subseteq X having continuous extensions to a Tychonoff space XX, with pointwise topology inherited from Cp(Y)C_{p}(Y). We recently proved Cp(Y)C_{p}(Y) is distinguished ⇔\Leftrightarrow it is a large subspace of RY\mathbb{R}^{Y}. We prove Cp(Y∣X)C_{p}\left( Y|X\right) is always a large subspace of Cp(Y)C_{p}(Y). Thus Cp(Y∣X)C_{p}\left( Y|X\right) is always quasibarrelled; always has a feral strong dual; has a quasibarrelled countable enlargement ⇔\Leftrightarrow YY is infinite; is distinguished ⇔\Leftrightarrow Cp(Y)C_{p}(Y) is distinguished; is a Montel space ⇔\Leftrightarrow YY is discrete and CC-embedded in XX. `Nice' countable covers for Cp(Y∣X)C_{p}\left( Y|X\right) yield potent summary theorems that solve open problems, characterize PP-spaces anew, and complete the list of Velichko variations. For example, Summary III: Assume YY is dense in XX. YY is a PP-space, or XX is pseudocompact, or both ⇔\Leftrightarrow Cp(Y∣X)C_{p}\left( Y|X\right) is countably covered by sets that are, respectively, relatively sequentially complete in Cp(Y)C_{p}(Y), or bounded, or both. Putting Y=XY=X, one quickly comprehends Velichko variations à la Arkhangel'skiĭ.

Author information

Contact details are reproduced from the original publication and may be historical.

Juan Carlos Ferrando

Centro de Investigación Operativa, Universidad Miguel Hernández, Elche, Spain

jc.ferrando@umh.es

Stephen A. Saxon

Dept. of Mathematics, University of Florida, Gainesville, U.S.A.

Keywords

  • Large subspace
  • distinguished
  • Montel
  • quasibarrelled
  • Cp(X)
  • C-embedded

Mathematics Subject Classification

46A03, 46A08, 46E10, 54C35

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Suggested citation

J. C. Ferrando, S. A. Saxon. “The Ever Large Subspace C_p(Y|X): Distinguished, Montel, Covered Nicely?.” Journal of Convex Analysis 33 (2026), No. 1&2, 361–375. https://doi.org/10.68381/jca33022

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Published by Heldermann Verlag, 2026. Rights now held by Banach Press.