Is the free locally convex space $L(X)$ nuclear?

Arkady Leiderman, Vladimir Uspenskij

Published 2021-06-25Version 1

Let $X$ be a Tychonoff space containing an infinite compact subset, we observe that the free locally convex space $L(X)$ is not nuclear; moreover, we prove a stronger result: $L(X)$ cannot be embedded in a product of Hilbert spaces. If $X$ is a $k$-space, then $L(X)$ is nuclear if and only if $X$ is countable and discrete. On the other hand, we show that $L(X)$ is nuclear for every projectively countable $P$-space (in particular, for every Lindel\"of $P$-space) $X$. In a sharp contrast with the nuclear property, we prove that $L(X)$ can be embedded in a product of reflexive Banach spaces, for every compact space $X$. However, we provide examples of completely metrizable $X$ such that $L(X)$ cannot be embedded in a product of reflexive Banach spaces.