I was asked to prove the next claim:
let $(V,\|\cdot\|)$ be a normed vector space, and let $U \subseteq V$ be an open and bounded set. Prove: $\partial U \neq \varnothing$. (where $\partial U$ denotes the boundary of $U$).
I've been trying to prove this for some time but with no success.
I also know this claim is false in a general metric space, but cant seem to understand why taking $V$ to be a normed space makes this true.
Hints and suggestions will be highly appreciated!