Show that E\H (Hyperplane) is arc-connected $\Longleftrightarrow$ H isn't a closed subspace Good morning,
Let $E$ be a real normed vector space and $H$ a hyperplane of $E$
Show that E\H is arc-connected $\Longleftrightarrow$ H isn't a closed subspace
I have no idea to solve it. But If $f$ a non-zero linear form such that $H=Ker(f)$ we have f(E\H)=$\mathbb{R^*}$ which is not arc-connected..
Edit: Thanks to @Martín-Blas Pérez Pinilla I have succeeded to prove E\H is arc-connected ⇒ H isn't a closed subspace
If someone can enlighten for the second implication,
Thank you in advance.

This problem only makes sense in infinite dimension. Idea: $H=\ker f$ is closed iff $f$ is continuous. And a discontinuous linear function is very discontinuous.
