Let $F$ be the space of real-valued sequences with only finitely many nonzero terms.

(a) For any $p \in[1, \infty)$, show that $F$ is dense in $\ell^{p}$. Is $F$ dense in $\ell^{\infty} ?$ Justify your answer.

(b) Let $p \in[1, \infty)$, and let $T: F \rightarrow F$ be an operator that is bounded in the $\|\cdot\|_{p}$-norm, i.e., there exists a $C$ such that $\|T x\|_{p} \leqslant C\|x\|_{p}$ for all $x \in F$. Show that there is a unique bounded operator $\widetilde{T}: \ell^{p} \rightarrow \ell^{p}$ satisfying $\left.\widetilde{T}\right|_{F}=T$, and that $\|\widetilde{T}\|_{p} \leqslant C$.

(c) For each $p \in[1, \infty]$ and for each $i=1, \ldots, 5$ determine if there is a bounded operator from $\ell^{p}$ to $\ell^{p}$ (in the $\|\cdot\|_{p}$ norm) whose restriction to $F$ is given by $T_{i}$ :

$$\begin{gathered} \left(T_{1} x\right)_{n}=n x_{n}, \quad\left(T_{2} x\right)_{n}=n\left(x_{n}-x_{n+1}\right), \quad\left(T_{3} x\right)_{n}=\frac{x_{n}}{n}, \\ \left(T_{4} x\right)_{n}=\frac{x_{1}}{n^{1 / 2}}, \quad\left(T_{5} x\right)_{n}=\frac{\sum_{j=1}^{n} x_{j}}{2^{n}} \end{gathered}$$

(d) Let $X$ be a normed vector space such that the closed unit ball $\overline{B_{1}(0)}$ is compact. Prove that $X$ is finite dimensional.