a) State and prove the Banach-Steinhaus Theorem.

[You may use the Baire Category Theorem without proving it.]

b) Let $X$ be a (complex) normed space and $S \subset X$. Prove that if $\{f(x): x \in S\}$ is a bounded set in $\mathbb{C}$ for every linear functional $f \in X^{*}$ then there exists $K \geqslant 0$ such that $\|x\| \leqslant K$ for all $x \in S .$

[You may use here the following consequence of the Hahn-Banach Theorem without proving it: for a given $x \in X$, there exists $f \in X^{*}$ with $\|f\|=1$ and $|f(x)|=\|x\|$.]

c) Conclude that if two norms $\|\cdot\|_{1}$ and $\|\cdot\|_{2}$ on a (complex) vector space $V$ are not equivalent, there exists a linear functional $f: V \rightarrow \mathbb{C}$ which is continuous with respect to one of the two norms, and discontinuous with respect to the other.