Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 358 / 2 / b

LetThe tail spaces decrease, so $C_{n+1}\subseteq C_n$. If $\lambda \in {\bigcap }_n{C}_n$, choose a unit vector $v_n$ supported after coordinate $n$ with $|⟨A{v}_n,v_n⟩-\lambda |<1/n$. Such vectors converge weakly to zero, hence $\lambda \in W_e(A)$.

Conversely, if $v_j\rightharpoonup 0$ and $⟨A{v}_j,v_j⟩\to \lambda$, then for every fixed $n$ the first $n$ coordinates of $v_j$ tend to zero. After normalizing $Q_n{v}_j$, its numerical values still tend to $\lambda$, so $\lambda \in C_n$. Therefore

When the intersection is nonempty, decreasing closed sets have distance functions increasing pointwise to the distance from their intersection; on each compact set this convergence is uniform. This is preciselyIf $W_e(A)=\varnothing$ and a compact $K$ met every $C_n$, nestedness and compactness would supply a convergent sequence whose limit belongs to all $C_n$, a contradiction. Hence $K\cap C_n=\varnothing$ for all sufficiently large $n$.

Solved by gpt-5.6-sol high.