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

If $f\notin K_{\epsilon }^{\prime }$, some relative weak-star neighbourhood $U$ of $f$ in $K$ has $\mathrm{diam}U\le \epsilon$. Every $g\in U$ has that same $U$ as a neighbourhood of norm diameter at most $\epsilon$, so $g\notin K_{\epsilon }^{\prime }$. Hence $K\setminus K_{\epsilon }^{\prime }$ is relatively weak-star open and the Szlenk derivation $K_{\epsilon }^{\prime }$ is weak-star closed in $K$.

Solved by gpt-5.6-sol high.

The set $K$ is weak-star compact by the Banach-Alaoglu theorem and weak-star metrizable by part a. Choose a decreasing countable neighbourhood base $(U_n)$ at $f$. Since $f\in K_{\epsilon }^{\prime }$, the set $U_n\cap K$ cannot lie inside the closed norm ball of radius $\epsilon /2$ about $f$, for then its diameter would be at most $\epsilon$. ChooseThe neighbourhood-base property gives $g_n\stackrel{w^{\ast }}{\to }f$.

Solved by gpt-5.6-sol high.

Start with the sequence from part ii. For each $n$, the definition of the operator norm gives $x_n\in B_X$ withBecause $g_n\stackrel{w^{\ast }}{\to }f$, pass recursively to a subsequence so far out that its $n$th member also satisfies

Part b supplies a weakly Cauchy subsequence $(x_{k_j})$. PutThen $(y_j)$ is weakly null, $h_j\stackrel{w^{\ast }}{\to }f$, and the two preceding estimates giveSince $f(y_j)\to 0$, discard finitely many terms to obtain $|h_j(y_j)|>\epsilon /16$ for every remaining $j$. Relabelling proves the claim.

Solved by gpt-5.6-sol high.

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.