Let Uinfty be a compact space, and let Ur be an increasing family of compact subspaces whose closure is all of Uinfty. That is, UrsubseteqUr′ if rler′ and Uinfty=overlinebigcupUr.
For rin[1,infty], let Yr=C(Ur,mathbbR) be the Banach space of real-valued continuous functions over Ur with the supremum norm. For rler′, let phir,r′:Yr′toYr be the restriction maps, so that Yinfty is the inverse limit of the spaces Yr. Write phir:YinftytoYr for the restriction map phir,infty.
Suppose there exists a family of continuous linear operators mr:YrtoYinfty such that |mr|leM for all r, and phircircmr is the identity map on Yr.
Question: Suppose GammasubseteqYinfty is compact. Does mrcircphir converge strongly to the identity operator on Gamma? That is, for all epsilon>0, does there exist R>0 such that if rgeR, then supyinGammaleft|(mrcircphir)(y)−yright|Yinfty<epsilon?
No comments:
Post a Comment