**"When is a locally convex space Eberlein-Grothendieck?"**
#### Leiderman, ArkadyWe 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 to be presented are: (1) For a first-countable space $X$ (in particular, for metrizable $X$) the space
$(C_{k}(X), w)$ is Eberlein-Grothendieck if and only if $X$ is both $\sigma$-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.
This is joint work with Jerzy Kakol. |