(a) Let $X$ be a normed vector space and $Y \subset X$ a closed subspace with $Y \neq X$. Show that $Y$ is nowhere dense in $X$.

(b) State any version of the Baire Category theorem.

(c) Let $X$ be an infinite-dimensional Banach space. Show that $X$ cannot have a countable algebraic basis, i.e. there is no countable subset $\left(x_{k}\right)_{k \in \mathbb{N}} \subset X$ such that every $x \in X$ can be written as a finite linear combination of elements of $\left(x_{k}\right)$.