(a) State and prove the Baire Category theorem.

Let $p>1$. Apply the Baire Category theorem to show that $\bigcup_{1 \leqslant q<p} l_{q} \neq l_{p}$. Give an explicit element of $l_{p} \backslash \bigcup_{1 \leqslant q<p} l_{q}$.

(b) Use the Baire Category theorem to prove that $C([0,1])$ contains a function which is nowhere differentiable.

(c) Let $(X,\|\cdot\|)$ be a real Banach space. Verify that the map sending $x$ to the function $e_{x}: \phi \mapsto \phi(x)$ is a continuous linear map of $X$ into $\left(X^{*}\right)^{*}$ where $X^{*}$ denotes the dual space of $(X,\|\cdot\|)$. Taking for granted the fact that this map is an isometry regardless of the norm on $X$, prove that if $\|\cdot\|^{\prime}$ is another norm on the vector space $X$ which is not equivalent to $\|\cdot\|$, then there is a linear function $\psi: X \rightarrow \mathbb{R}$ which is continuous with respect to one of the two norms $\|\cdot\|,\|\cdot\|^{\prime}$ and not continuous with respect to the other.