(a) Let $X$ be a Banach space and consider the open unit ball $B=\{x \in X:\|x\|<1\}$. Let $T: X \rightarrow X$ be a bounded operator. Prove that $\overline{T(B)} \supset B \operatorname{implies} T(B) \supset B$.

(b) Let $P$ be the vector space of all polynomials in one variable with real coefficients. Let $\|\cdot\|$ be any norm on $P$. Show that $(P,\|\cdot\|)$ is not complete.

(c) Let $f: \mathbb{C} \rightarrow \mathbb{C}$ be entire, and assume that for every $z \in \mathbb{C}$ there is $n$ such that $f^{(n)}(z)=0$ where $f^{(n)}$ is the $n$-th derivative of $f$. Prove that $f$ is a polynomial.

[You may use that an entire function vanishing on an open subset of $\mathbb{C}$ must vanish everywhere.]

(d) A Banach space $X$ is said to be uniformly convex if for every $\varepsilon \in(0,2]$ there is $\delta>0$ such that for all $x, y \in X$ such that $\|x\|=\|y\|=1$ and $\|x-y\| \geqslant \varepsilon$, one has $\|(x+y) / 2\| \leqslant 1-\delta$. Prove that $\ell^{2}$ is uniformly convex.