0

Define a hyperplane $H$ in a linear normed vector space $X$ to be a maximal proper linear variety. That is, $H\ne X$ is a linear variety, and for any linear variety $V\supset H$, either $V=X$ or $V=H$. Then, $\bar{H}=H$ or $X$.

I know that $H$ can be written as $H=\{f(x)=c\}$ for some linear functional $f$. I am not sure when $\bar{H}=X$ and how to prove that if $H$ is not closed then $\bar{H}=X$.

I know that $H=\bar{H}$ iff $f$ is continuous, so it has to be the case that $f$ is discontinuous. But I don't know how to use this to show the result.

keepfrog
  • 664
  • It is well known (and proved earlier on MSE) that $H$ is closed if $f$ is continuous and dense if it is not continuous. – Kavi Rama Murthy Jan 22 '24 at 04:55
  • Could you kindly point out to the reference? I know that $H$ is closed if and only if $f$ is continuous. What I struggle is the fact that if $H$ is not closed then $\bar{H}=X$. – keepfrog Jan 22 '24 at 05:10
  • https://math.stackexchange.com/questions/1255341/ker-f-is-either-dense-or-closed-when-f-is-a-linear-functional-on-a-normed-l?rq=1 – Kavi Rama Murthy Jan 22 '24 at 05:13
  • 1
    Your definition of a hyperplane makes the proof of the non-closed case almost trivial: for any linear subspace $U$ of a normed space the closure is again a linear subspace by the linearity of the limit operator. – Hagen Knaf Jan 22 '24 at 06:21

0 Answers0