Let $E$ be a not normable locally convex space, define $$F: E'\times E\to \mathbb R$$ $$(f,e)\to f(e)$$ I have to show that $F$ is not continuous when $E'\times E$ is given product topology.
I was reading an article and i came across with this fact.. Please give me atleast a hint to start..
My try: I know that $E$ is normable if and only if origin has a convex bounded neighborhood. So i was trying to produce any such neighborhood to contradict to assumption. Assume $F$ is continuous, then we have
$\{(f,e): a<f(e)<b\}$ is open in product topology of $E'\times E$, for any $a,b\in \mathbb R$. This means there is some open set $U'$ in $E'$ and $U$ in $E$ such that
$$U'\times U\subset \{(f,e): a<f(e)<b\}$$
Now let
$V:=\{e\in E: a<f(e)<b;\forall f\in U'\}$, this is open convex neighborhood of origin, but how to prove this is bounded. Or we have any other way to produce such a neighborhood.
Thanks for your time.