root systems in Euclidean space

Question

Let $E$ be a Euclidean space and suppose $\mathbb{\Phi}$ is a root system https://en.wikipedia.org/wiki/Root_system. Can anyone show why $E\setminus\cup_{\alpha}P_\alpha$ is nonempty? $P_\alpha$ is the hyperplane perpendicular to $\alpha.$

Answer 1

As Jyrki Lahtonen showed in

https://math.stackexchange.com/questions/60698/

and also proved by Pete L. Clark in the note

$\;\;\;\;\;\;\;$http://alpha.math.uga.edu/~pete/coveringnumbersv2.pdf

we have the result:

If $F$ is an infinite field, and $V$ is a finite dimensional vector space over $F$, then $V$ is not a finite union of proper subspaces.

Applying the above result to the problem at hand, since $\mathbb{\Phi}$ is finite, and each $P_\alpha$ is a proper subspace of $E$, we can't have $$E=\bigcup_{\alpha\in{\large{\Phi}}}P_\alpha$$ hence $E{\,\setminus}\left(\bigcup_{\alpha\in{\large{\Phi}}}P_\alpha\right)\ne{\large{\varnothing}}$,$\;$as required.