= Solution
If $x_i\rightharpoonup x$ and $\psi\in Y^*$, then $\psi\circ T\in X^*$ because $T$ is bounded and linear. Therefore $\psi(Tx_i)\to\psi(Tx)$, proving weak continuity of $T$.
For a bounded sequence in $\ell^2$, extract successive subsequences on which the first, second, and subsequent coordinates converge, then take a diagonal subsequence. Call the coordinate limits $x_j$. Every finite partial sum satisfies $\sum_{j\le m}|x_j|^2\le1$, so $x\in\ell^2$ and $\|x\|\le1$. For any $h\in\ell^2$, the pairing on its first $m$ coordinates converges, while the pairing on the tail is bounded uniformly by $2\|h_{>m}\|_2$. First choose $m$ large, then take the subsequence limit. The <Riesz representation theorem> identifies these pairings with every <continuous linear functional>, proving the requested <weak convergence>.
For $\ell^1$, <norm convergence> implies <weak convergence> immediately. Conversely, suppose $z_i\rightharpoonup0$ but a subsequence has $\|z_i\|_1\ge\delta>0$. Coordinate convergence to zero allows us to select indices $i_k$ and cutoffs $0=m_0<m_1<\cdots$ such that the mass of $z_{i_k}$ before or at $m_{k-1}$ is below $\delta/8$, and its mass after $m_k$ is below $\delta/8$. Its mass on the intervening block is then at least $3\delta/4$. On that block set $a_j=\overline{z_{i_k,j}}/|z_{i_k,j}|$ when nonzero and zero otherwise. These prescriptions define $a\in\ell^\infty$. The bounded <linear functional> $z\mapsto\sum_ja_jz_j$ has real part at least $3\delta/4-\delta/4=\delta/2$ on every selected $z_{i_k}$, contradicting <weak convergence>. Translating by the limit proves \b[weak and <norm convergence> of sequences coincide in $\ell^1$], the <Schur property>.
A <compact operator> sends bounded sets to relatively norm-compact sets. Every bounded sequence in $\ell^2$ has a weakly convergent subsequence by the argument above, after scaling into the unit ball. Its image under $T:\ell^2\to\ell^1$ converges weakly and therefore in norm by the <Schur property>. Sequential compactness in a metric space gives relative compactness of the image of the unit ball. \b[Every <bounded linear operator> from $\ell^2$ to $\ell^1$ is compact.]
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