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Best approximation in Orlicz spaces. (English) Zbl 0746.41027

Summary: Let \(X\) be a real Banach space and \((\Omega,\mu)\) be a finite measure space and \(\phi\) be a strictly increasing convex continuous function on \([0,\infty)\) with \(\phi(0)=0\). The space \(L_ \phi(\mu,X)\) is the set of all measurable functions \(f\) with values in \(X\) such that \(\int_ \Omega \phi(\| c^{-1}f(t)\|)d\mu(t)<\infty\) for some \(c>0\). One of the main results of this paper is: “For a closed subspace \(Y\) of \(X\), \(L_ \phi(\mu,Y)\) is proximinal in \(L_ \phi(\mu,X)\) if and only if \(L^ 1(\mu,Y)\) is proximinal in \(L^ 1(\mu,X)\).” As a result if \(Y\) is reflexive subspace of \(X\), then \(L_ \phi(\phi,Y)\) is proximinal in \(L_ \phi(\mu,X)\). Other results on proximinality of subspaces of \(L_ \phi(mu,x)\) are proved.

MSC:

41A50 Best approximation, Chebyshev systems
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
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