论文标题
$ω^ω$ - 基本和无限维的紧凑型集合在本地凸空间中
$ω^ω$-Base and infinite-dimensional compact sets in locally convex spaces
论文作者
论文摘要
如果$ e $具有$ω^ω$ - base，则在本地凸空间（LCS）$ e $具有$ω^ω$ - 基本，如果$ e $具有零件$ \ {u_α：α\inΩ^ω\} $的零，以至于$u_β\u_β\ subsetequ_α$ for $leqβ$。具有$ω^ω$ - 基本的LCS类很大，其中包含所有$（LM）$ - 空间（因此$（LF）$ - 空间），强大的fréchetLCS（因此分布空间$ d'（ω）$）的强大双重。 Cascales-Orihuela的显着结果指出，具有$ω^ω$ - 基本的LCS中的每个紧凑型都可以Metrizable。我们的主要结果表明，每个具有$ω^ω$ - 基本的无数尺寸LCS都包含一个无限尺寸的可衡量的紧凑型子集。另一方面，赋予本地最好的凸形拓扑的可计数空间$φ$具有$ω^ω$ - 基本，但不包含无限维的紧凑型子集。事实证明，$φ$是一个独特的无限维当地凸空间，它是$ k _ {\ mathbb {r}} $ - 不包含无限维紧凑型子集的空间。提供了空格的应用程序$ C_ {P}（X）$。
A locally convex space (lcs) $E$ is said to have an $ω^ω$-base if $E$ has a neighborhood base $\{U_α:α\inω^ω\}$ at zero such that $U_β\subseteq U_α$ for all $α\leqβ$. The class of lcs with an $ω^ω$-base is large, among others contains all $(LM)$-spaces (hence $(LF)$-spaces), strong duals of distinguished Fréchet lcs (hence spaces of distributions $D'(Ω)$). A remarkable result of Cascales-Orihuela states that every compact set in a lcs with an $ω^ω$-base is metrizable. Our main result shows that every uncountable-dimensional lcs with an $ω^ω$-base contains an infinite-dimensional metrizable compact subset. On the other hand, the countable-dimensional space $φ$ endowed with the finest locally convex topology has an $ω^ω$-base but contains no infinite-dimensional compact subsets. It turns out that $φ$ is a unique infinite-dimensional locally convex space which is a $k_{\mathbb{R}}$-space containing no infinite-dimensional compact subsets. Applications to spaces $C_{p}(X)$ are provided.
