I am trying to solve this little problem.
Suppose you have a normed vector space $E$. Let $H$ be a hyperplane ( $H=\{x\in E: f(x)= \alpha\}$ for some linear functional $f$ and some real number $\alpha$) and let $V$ be an affine subspace (i.e. $V=U+a$ where $a \in E$ and $U$ is a vector subspace of $E$) that contains $H$.
In my previous post I proved the following: either $H=V$ or $V=E$.
Check it out here: A problem involving a hyperplane and an affine subspace
Now, I want to deduce from this that H is either closed or dense in $E$. Any thoughts?