On the difficulty of preserving monotonicity via projections and related results

• Douglas Mupasiri ^[1] ; Michael P. Prophet ^[1]
  1. [1] University of Northern Iowa

     University of Northern Iowa

     City of Cedar Falls, Estados Unidos

     
• Localización: Jaen journal on approximation, ISSN 1889-3066, ISSN-e 1989-7251, Vol. 2, Nº. 1, 2010, págs. 1-12
• Idioma: inglés
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• Resumen

  • A subspace V of a Banach space X is said to be complemented if there exists a (bounded) projection mapping X onto V . Obviously all subspaces of finitedimension are complemented. The goal of this note is to show that there are (relatively) few monotonically complemented subspaces of finite-dimension in X = (C[a, b], ·∞); that is, finite-dimensional subspaces V ⊂ X for which there exists a projection P : X → V such that P f is monotone-increasing whenever f is. We obtain several corollaries from this consideration, including a result describing the difficulty of preserving n-convexity via a projection.