(a) Prove that in a finite-dimensional normed vector space the weak and strong topologies coincide.

(b) Prove that in a normed vector space $X$, a weakly convergent sequence is bounded. [Any form of the Banach-Steinhaus theorem may be used, as long as you state it clearly.]

(c) Let $\ell^{1}$ be the space of real-valued absolutely summable sequences. Suppose $\left(a^{k}\right)$ is a weakly convergent sequence in $\ell^{1}$ which does not converge strongly. Show there is a constant $\varepsilon>0$ and a sequence $\left(x^{k}\right)$ in $\ell^{1}$ which satisfies $x^{k} \rightarrow 0$ and $\left\|x^{k}\right\|_{\ell^{1}} \geqslant \varepsilon$ for all $k \geqslant 1$.

With $\left(x^{k}\right)$ as above, show there is some $y \in \ell^{\infty}$ and a subsequence $\left(x^{k_{n}}\right)$ of $\left(x^{k}\right)$ with $\left\langle x^{k_{n}}, y\right\rangle \geqslant \varepsilon / 3$ for all $n$. Deduce that every weakly convergent sequence in $\ell^{1}$ is strongly convergent.

[Hint: Define $y$ so that $y_{i}=\operatorname{sign} x_{i}^{k_{n}}$ for $b_{n-1}<i \leqslant b_{n}$, where the sequence of integers $b_{n}$ should be defined inductively along with $\left.x^{k_{n}} .\right]$

(d) Is the conclusion of part (c) still true if we replace $\ell^{1}$ by $L^{1}([0,2 \pi]) ?$