Journal of Convex Analysis 33 (2026), No. 1&2, 361--375
Copyright Heldermann Verlag 2026
The Ever Large Subspace Cp(Y|X): Distinguished, Montel, Covered Nicely?
Juan Carlos Ferrando
Centro de Investigacion Operativa, Universidad Miguel Hernandez, Elche, Spain
jc.ferrando@umh.es
Stephen A. Saxon
Dept. of Mathematics, University of Florida, Gainesville, U.S.A.
[Abstract-pdf]
$C_{p}\left( Y|X\right) $ denotes the real-valued continuous functions on $Y\subseteq X$ having continuous extensions to a Tychonoff space $X$, with pointwise topology inherited from $C_{p}(Y)$. We recently proved $C_{p}(Y)$\emph{\ is distinguished }$\Leftrightarrow$ \emph{it is a large subspace of} $\mathbb{R}^{Y}$. We prove $C_{p}\left( Y|X\right) $\emph{\ is always a large subspace of }$C_{p}(Y)$% \emph{.} Thus $C_{p}\left( Y|X\right) $\emph{\ is always quasibarrelled; always has a feral strong dual; has a quasibarrelled countable enlargement }$% \Leftrightarrow $\emph{\ }$Y$\emph{\ is infinite; is distinguished }$% \Leftrightarrow $\emph{\ }$C_{p}(Y)$\emph{\ is distinguished; is a Montel space }$\Leftrightarrow $\emph{\ }$Y$\emph{\ is discrete and }$C$% \emph{-embedded in }$X$. `Nice' countable covers for $C_{p}\left( Y|X\right) $ yield potent summary theorems that solve open problems, characterize $P$% -spaces anew, and complete the list of Velichko variations. For example, Summary III: \emph{Assume }$Y$\emph{\ is dense in }$X$\emph{.} $Y$\emph{\ is a }$P$\emph{-space, or }$X$\emph{\ is pseudocompact, or both }$% \Leftrightarrow $\emph{\ }$C_{p}\left( Y|X\right) $\emph{\ is countably covered by sets that are, respectively, relatively sequentially complete in }% $C_{p}(Y)$,\emph{\ or bounded, or both}. Putting $Y=X$, one quickly comprehends Velichko variations \textit{\`a la} Arkhangel'ski\u{\i}.
Keywords: Large subspace, distinguished, Montel, quasibarrelled, $C_{p}(X)$, $C$-embedded.
MSC: 46A03, 46A08, 46E10, 54C35.
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