Is the relative interior of a subspace which is not closed empty?

Question

In a general Banach space, the relative interior of a linear subspace which is not closed is empty, why ?

Answer 1

Let $U$ be a subspace of a Banach space $V$. Suppose $x\in U$ is an interior point of $U$, which means that $$ B(x,\varepsilon)=\{y:\|x-y\|<\varepsilon\}\subseteq U $$ for some $\varepsilon>0$. Since $U$ is a subspace, also $$ B(0,\varepsilon)=-x+B(x,\varepsilon) $$ is contained in $U$. Now, for every $v\in V$, $v\ne0$, the vector $$ \frac{1}{2\varepsilon\|v\|}v\in B(0,\varepsilon) $$ which means that $U$ contains a scalar multiple of every nonzero vector. Therefore $U=V$.

So the only subspace of $V$ that has interior points is $V$ itself (provided $V\ne\{0\}$).