Is an open bounded subset of $\mathbb{R}^n$ a Banach space?

Question

Let $\Omega$ be an open bounded subset of $\mathbb{R}^n$. Is $\Omega$ a Banach (sub)space?

Answer 1

Subspaces of $\mathbf R^n$ are all unbounded and closed.

Answer 2

A Banach subspace is a linear subspace. Part of the definition of that is being closed under multiplication by scalars, i.e. if $x \in S$ then $tx \in S$ for all scalars $t$. Since $\|tx\| = |t| \|x\|$, the only bounded set that satisfies this is $\{0\}$. But $\{0\}$ is not open.