If not distinguished, is $C_{p}\left ( X\right )$ even close?

Abstract:

$C_{p}\left ( X\right )$ is distinguished $\Leftrightarrow$ the strong dual $L_{\beta }\left ( X\right )$ is barrelled $\Leftrightarrow$ the strong bidual $M\left ( X\right ) =\mathbb {R}^{X}$. So one may judge how nearly distinguished $C_{p}\left ( X\right )$ is by how nearly barrelled $L_{\beta }\left ( X\right )$ is, and also by how near the dense subspace $M\left ( X\right )$ is to the Baire space $\mathbb {R}^{X}$. Being Baire-like, $M\left ( X\right )$ is always fairly close to $\mathbb {R}^{X}$ in that sense. But if $C_{p}\left ( X\right )$ is not distinguished, we show the codimension of $M\left ( X\right )$ is uncountable, i.e., $M\left ( X\right )$ is algebraically far from $\mathbb {R}^{X}$, and moreover, $L_{\beta }\left ( X\right )$ is very far from barrelled, not even primitive. Thus we profile weak barrelledness for $L_{\beta }\left ( X\right )$ and $M\left ( X\right )$ spaces. At the same time, we characterize those Tychonoff spaces $X$ for which $C_{p}\left ( X\right )$ is distinguished, solving the original problem from our series of papers.