Strong uniqueness and Lipschitz continuity of metric projections: A generalization of the classical Haar theory (Q1120091)

Let T be a locally compact Hausdorff space and let \(C_ 0(T)\) be the Banach space of real-valued continuous functions f on T which vanish at infinity, with the supremum norm. For two subsets A, B in \(C_ 0(T)\), define \(d(A,B)=\sup_{f\in A}\inf_{g\in B}\| f-g\|\) and \(D(A,B)=\max \{d(A,B),d(B,A)\}\). If G is a subspace of \(C_ 0(T)\), the metric projection \(P_ G\) from \(C_ 0(T)\) to G is defined as \(P_ G(f)=\{f\in G:\| f-g\| =d(f,G)\}\) for all \(f\in C_ 0(T)\). Then, \(P_ G(f)\) is Hausdorff strongly unique if there exists \(r(f)>0\) such that \(\| f-g\| \geq d(f,G)+r(f)d(g,P_ G(f))\) for \(g\in G\); \(P_ G\) is Hausdorff continuous at f if \(\lim_{\epsilon \to 0^+}\sup_{\| g-g\| \leq \epsilon}D(P_ G(f),\) \(P_ G(h))=0\); and \(P_ G\) is Hausdorff-Lipschitz continuous at f if there exists \(s(f)>0\) such that \(D(P_ G(f),P_ G(h))\leq s(f)\| f-h\|\) for all \(h\in C_ 0(T)\). The main results of this paper can be summarized as follows. Suppose that G is a finite-dimensional subspace of \(C_ 0(T)\). Then the following statements are mutually equivalent: (1) for every nonzero \(g\in G\), \(card(bdZ(g))\leq \dim \{p\in G:int Z(g)\subset Z(p)\}- 1;\) (2) \(P_ G\) is Hausdorff continuous at every \(f\in C_ 0(T)\); (3) \(P_ G(f)\) is Hausdorff strongly unique for all \(f\in C_ 0(T)\); (4) \(P_ G\) is Hausdorff-Lipschitz continuous at every \(f\in C_ 0(T)\). If T contains no isolated points, then all above are equivalent to (5) \(SU_ G=U_ G\). G satisfies the Haar condition.