Absolute projection constant of the linear functions in a Lebesgue space (Q1104506)

If E and X are Banach spaces such that \(E\subset X\) then the relative projection constant of E with respect to X is defined by \(\lambda (E;X)=\inf \| P\|\) where the infimum is taken over all linear projections P of X onto E. The absolute projection constant of E is defined by \(\lambda (E)=\sup_{x}\lambda (E;X).\) \textit{C. Franchetti} and \textit{E. W. Cheney} [Duke Math. J. 43, 501-510 (1976; Zbl 0335.41021)] computed the relative projection constant \(\lambda(V;L^ 1[a,b])\), where V is the subspace of \(L^ 1[a,b]\) formed of all linear functions. The aim of this note is to show that \(\lambda(V;L^ 1[a,b])= \lambda(V).\) To this end, the author shows that the subspace \(W=sp\{t,1-t^ 2\}\) of \(C=C[-1,1]\) is linearly isometric to V, finds a projection \(P: C\to W\) such that \(\| P\| =\lambda (V;L^ 1)\) and then uses the well-known relations: \(\lambda (W;C)=\lambda (W)\) and \(\lambda (V)=\lambda (W)\).