The prevalence of strong uniqueness in \(L^ 1\) (Q1108516)

Let (X,\(\Sigma\),\(\mu)\) be a positive \(\sigma\)-finite measure space and let M be a finite dimensional subspace of \(L^ 1\equiv L^ 1(X,\Sigma,\mu)\). Define \(U_ M:=\{f\in L^ 1:f\) has a unique best approximation from \(M\}\) and \(SU_ M:=\{f\in L^ 1:f\) has a strongly unique best approximation from \(M\}\). The main results shown here are that \(SU_ M\) is dense in \(U_ M\), if M is almost Chebyshev then \(SU_ M\) is dense in \(L^ 1\), and if \(L^ 1\) is finite dimensional then \(SU_ M=U_ M\). Further it is shown that if in the case (X,\(\Sigma\),\(\mu)\) is nonatomic, in which case M is almost Chebyshev, then \(SU_ M\) is a dense set of first category.