Let $E$ be a locally convex space, let $U$ be a convex balanced closed neighbourhood of $0$ in $E$. Let $F=\bigcap_{n\in\mathbf{N}}n^{-1}U$, which is a closed linear subspace of $E$. In fact $F$ is the kernel of the Minkowski semi-norm of $U$. This semi-norm induces a norm on the vector space $E/F$.
Is the described topology on $E/F$ the quotient topology?
The image of $U$ in the quotient topology is a neighbourhood of $0$, which does not contain a subspace. In order to prove the question we need to show that this image is bounded. So does any unbounded balanced convex neighbourhood of $0$ contain a subspace?