Let $\mathbb{RP^{n-1}}=\mathbb{R^n}/ \sim $ where x ~ y iff $\exists \ \lambda \in \mathbb{R} \ s.t \ \lambda x=y$
Can $\mathbb{RP^{n}}$ be given the structure of an $\mathbb{R}$-module . Moreover, can $\mathbb{RP^{n}}$ be given a structure of a normed linear space.
Since $\mathbb R$ is a field, an $\mathbb R$-module would be an $\mathbb R$ vector space. Any $n$-dimensional $\mathbb R$ vector space is isomorphic to $\mathbb R^n$. And $\mathbb{RP}^n$ has a topology which differs from that of $\mathbb R^n$. Therefore there can be no isomorphism between these.
In your definition, you didn't exclude the null vector from the quotient, thus breaking transitivity of the equivalence relation. If you can allow for a similarly lax treatment of special cases, then you can omit one $\mathbb{RP}^{n-1}$ from $\mathbb{RP}^n$ and treat the rest as isomorphic to $\mathbb R^n$. The removed space would be the elements at infinity.