Problem. Let Z1⊇Z2⊇Z3⊇⋯ be a chain of closed convex subsets of a Hilbert space H, show that:Since Zn's are not subspace of H, PZn itself is not a linear map, and therefore standard tools from functional analysis cannot be used. Nevertheless, by assuming Zn's are further a closed subspace, it is instructive to see a weak-convergence argument to conclude the result in (a) --- a slightly complicated solution.
(a) If ⋂∞n=1Zn≠∅, then ‖x−PZnx‖→‖x−PZx‖.
(b) If ⋂∞n=1Zn=∅, then ‖x−PZnx‖→∞.
Some also asked me can ⋂∞n=1Zn=∅ happen? Below is an example to show there is a chain of closed convex subsets with empty intersection:
Example. Let Zn={(1,1√2,…,1√n,a1,a2,…):(a1,a2,…)∈ℓ2}, then Zn is convex, a closed subset of ℓ2, and descending in n. However, if there is (x1,x2,…)∈⋂∞n=1Zn, then necessarily xn=1/√n for every n, and this element is not in ℓ2, a contradiction. So ⋂∞n=1Zn=∅.
Next what is d(x,Zn) in this particular example? By definition for z∈Zn we have ‖x−z‖2=n∑k=1|xk−1√k|2+∑k>n|xk−zk|2, hence for fixed x∈ℓ2 the smallest possible ‖x−z‖ is d(x,Zn)=√n∑k=1|xk−1√k|2, of course this diverges to infinity as n→∞ for any fixed x∈ℓ2. What's more, PZnx=(1,1√2,…,1√n,xn+1,xn+2,…).◼
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