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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.

1 Answers1

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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.

Martín-Blas Pérez Pinilla
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  • Yes, in finite dimension I have shown that it does not work (look at my notes: 'But..'). Thank you, with your idea I have succeeded to prove E\H is arc-connected $\Rightarrow$ H isn't a closed subspace –  Jan 25 '14 at 17:05