A Lindelöf space is a topological space in which every open cover has a countable subcover. 


1. Does there exists a Lindelöf locally convex space which is not second countable?
2. I am also looking for a separable locally convex which is not second countable!



Separable Lindelöf locally convex spaces that are not second-countable
A space of the form $C\_p(X)$ (the space of continuous functions on a Tychonoff space $X$, in the pointwise-topology (i.e. the subspace topology induced from the product topology on $\mathbb{R}^X$) is often an example; note that such spaces are locally convex spaces:


$C\_p(X)$ is separable when $X$ is second countable, but $C\_p(X)$ is not second countable unless $X$ is countable. $C\_p(X)$ is Lindelöf when $X$ is second countable and metrisable. (Proofs can be found in Arhangel'skij's book on $C\_p(X)$ or Tkachuk's series of books on them etc.) So concretely, $C\_p(\mathbb{R})$ or $C\_p([0,1])$ is an example of a Lindelöf, separable locally convex space that is not second countable. (It does have countable network weight, as $nw(X) = nw(C\_p(X))$ for all $X$). 


