Banach Space Theory and its Applications pp 44-54 | Cite as
Best approximation and intersections of balls
Abstract
For a linear subspace G of the normed linear space E and xεE, let PG(x) be the set of all best approximations of x out of G. Observing that for each x,yεE we always have dist (y,PG(x))≥‖x−y‖-dist(x,G), we study the subspaces G with the property — which we call property (*) — that this inequality is an equality for each xεE with PG(x)≠φ and each gεG. This property generalizes the notion of semi L-summand studied by A.Lima. For a subspace G with property (*), the one-sided Gateaux differential of the norm at xεE with OεPG(x), in the direction gεG equals the distance of −g to the cone spanned by PG(x). Using this result, we obtain a characterization of those xεE with OεPG(x) in order that the cone spanned by PG(x) to be norm-dense in G. When G is proximinal, property (*) is equivalent with 1 1/2-ball property studied by D.Yost. We give geometrical characterizations of the subspaces with property (*), as well as with 1 1/2-ball property.
Keywords
Banach Space Linear Subspace Closed Subspace Normed Linear Space Compact Hausdorff SpacePreview
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