If not distinguished, is 𝐶_{𝑝}(𝑋) even close?,Proceedings of the American Mathematical Society

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If not distinguished, is 𝐶_{𝑝}(𝑋) even close?
Proceedings of the American Mathematical Society ( IF 1 ) Pub Date : 2021-03-25 , DOI: 10.1090/proc/15439
J. C. Ferrando , Stephen A. Saxon

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

for which $C_{p}\left ( X\right )$ is distinguished, solving the original problem from our series of papers.

中文翻译：

如果不区分，𝐶_{𝑝}（𝑋）甚至会关闭吗？

摘要：尊贵的双重对桶的坚强的双strong 。因此，可以判断出近乎有区别的有多近乎桶形，以及有致密的子空间离Baire空间有多近。从某种意义上说，像贝儿一样，总是很接近。但是，如果不加以区分，则表明的余维是不可数的，即，在代数上远离，而且与桶形相距甚远，甚至不是原始的。因此，我们分析了和空间的弱桶状度。与此同时，我们描述了那些吉洪诺夫空间为这 $C_ {p} \ left（X \ right）$ $\ Leftrightarrow$ $L _ {\ beta} \ left（X \ right）$ $\ Leftrightarrow$ $M \ left（X \ right）= \ mathbb {R} ^ {X}$ $C_ {p} \ left（X \ right）$ $L _ {\ beta} \ left（X \ right）$ $M \左（X \ right）$ $\ mathbb {R} ^ {X}$ $M \左（X \ right）$ $\ mathbb {R} ^ {X}$ $C_ {p} \ left（X \ right）$ $M \左（X \ right）$ $M \左（X \ right）$ $\ mathbb {R} ^ {X}$ $L _ {\ beta} \ left（X \ right）$ $L _ {\ beta} \ left（X \ right）$ $M \左（X \ right）$

$C_ {p} \ left（X \ right）$ 卓著，解决了我们系列论文中的原始问题。

更新日期：2021-04-22

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