论文标题
何时在本地凸出空间eberlein grothendieck？
When is a locally convex space Eberlein-Grothendieck?
论文作者
论文摘要
在本文中，我们对这些本地凸空间$ e $进行系统研究，以使$（e，w）$是（线性）eberlein-grothendieck，其中$ w $是$ e $的弱拓扑。 令$ c_ {k}（x）$为tychonoff空间上连续实现功能的空间，并带有紧凑型开放式拓扑。我们论文的主要结果是：（1）对于首次计算空间$ x $（尤其是对于Metrizable $ x $），本地凸出空间$（c_ {k}（x），w），w）$是eberlein-grothendieck，并且仅当$ x $都是$σ$ -compact和compart; （2）$（c_ {k}（x），w）$是线性的eberlein-grothendieck，并且仅当$ x $紧凑时。 我们表征了$ e $，以便$（e，w）$是其他几个重要类别的本地凸空间$ e $的eberlein-grothendieck。 另外，我们表明，$（e，w）$的$ e $类是线性的eberlein-grothendieck，可以保留线性连续的商。提供了各种说明示例。
In this paper we undertake a systematic study of those locally convex spaces $E$ such that $(E, w)$ is (linearly) Eberlein-Grothendieck, where $w$ is the weak topology of $E$. Let $C_{k}(X)$ be the space of continuous real-valued functions on a Tychonoff space $X$ endowed with the compact-open topology. The main results of our paper are: (1) For a first-countable space $X$ (in particular, for a metrizable $X$) the locally convex space $(C_{k}(X), w)$ is Eberlein-Grothendieck if and only if $X$ is both $σ$-compact and locally compact; (2) $(C_{k}(X), w)$ is linearly Eberlein-Grothendieck if and only if $X$ is compact. We characterize $E$ such that $(E, w)$ is linearly Eberlein-Grothendieck for several other important classes of locally convex spaces $E$. Also, we show that the class of $E$ for which $(E, w)$ is linearly Eberlein-Grothendieck preserves linear continuous quotients. Various illustrating examples are provided.
