(a) State and prove the Baire category theorem.

(b) Let $X$ be a normed space. Show that every proper linear subspace $V \subset X$ has empty interior.

(c) Let $\mathcal{P}$ be the vector space of all real polynomials in one variable. Using the Baire category theorem and the result from (b), prove that for any norm $\|\cdot\|$ on $\mathcal{P}$, the normed space $(\mathcal{P},\|\cdot\|)$ is not a Banach space.