Past exam of the mathematics course of the University of Cambridge / 2006 / ii / Paper 3 / 21G / Solution

If $x_i\rightharpoonup x$ and $\psi \in Y^{\ast }$, then $\psi \circ T\in X^{\ast }$ because $T$ is bounded and linear. Therefore $\psi (T{x}_i)\to \psi (Tx)$, 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 ${\sum }_{j\le m}|x_j{|}^2\le 1$, so $x\in {\ell }^2$ and $\Vert x\Vert \le 1$. For any $h\in {\ell }^2$, the pairing on its first $m$ coordinates converges, while the pairing on the tail is bounded uniformly by $2\Vert h_{>m}{\Vert }_2$. 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 $z_i\rightharpoonup 0$ but a subsequence has $\Vert z_i{\Vert }_1\ge \delta >0$. 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 $z\mapsto {\sum }_j{a}_j{z}_j$ 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 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. Every bounded linear operator from ${\ell }^2$ to ${\ell }^1$ is compact.