Non-existence of Monotonically Complemented Subspaces of C[a, b] and Other Negative Results
نویسنده
چکیده
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 finite-dimension 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, finitedimensional subspaces V ⊂ X for which there exists a projection P : X → V such that Pf 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.
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