Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 106 / 2 / c / iv

Part i makes $K_{\epsilon }^{\prime }$ a closed subset of the compact metrizable space $K$, so it is compact metrizable and therefore separable. Choose the stated dense sequence $(f_m)$ and, using part iii, the sequences $(g_{m,n})$ and $(y_{m,n})$. In constructing the universal weakly null sequence $(z_n)$ from part b, retain a tail of each $y_m$ and write $z_{j(m,n)}=y_{m,n}$ along the retained subsequence, where $j(m,n)\to \infty$.

Eachis weak-star closed. Since $z_n\rightharpoonup 0$, every fixed $f\in K$ belongs to some $A_N$, so $K={\bigcup }_N{A}_N$. The compact Hausdorff space $K$ is a Baire space, and the Baire category theorem implies that some $A_N$ has nonempty relative weak-star interior.

Suppose $K_{\epsilon }^{\prime }=K$. Choose a nonempty relatively open $O\subseteq A_N$ and then $f_m\in O$ by density. Since $g_{m,n}\stackrel{w^{\ast }}{\to }{f}_m$, eventually $g_{m,n}\in O$. For a sufficiently late retained index, also $j(m,n)\ge N$, and hencecontradicting $|g_{m,n}(y_{m,n})|>\epsilon /16$. Therefore the One-step Szlenk derivation for a separable dual gives $K_{\epsilon }^{\prime }⊊K$ whenever $K$ is nonempty.

Solved by gpt-5.6-sol high.