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On Stability of the Metric Projection Operator

András Kroó, Allan Pinkus

2013
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SIAM Journal on Mathematical Analysis
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Let M be a closed linear subspace of a normed linear space X. For a given f ∈ X denote by P M f the set of best approximations to f from M . The operator P M is termed the metric projection onto M . In this paper we are interested in the stability of the metric projection P M relative to perturbations of the subspace M . We mainly consider the case where X = L p , p ∈ [1, ∞]. We consider a measure of distance d(M, N) between subspaces M and N and estimate P M f − P N f in terms of d(M, N) and f
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... ms of d(M, N) and f . Typically such an estimate will be of order d(M, N) β with some β which, in general, depends on the geometry of the space X. (pinkus@technion.ac.il). 639 640 ANDRAS KROÓ AND ALLAN PINKUS and Milman [10] . A somewhat similar concept was introduced in Gohberg and Markus [5], namely, d(M, N ) is the Hausdorff distance between the unit spheres of the linear subspaces M and N . Note that on the one hand d(M, N ) ≤ d(M, N ), while on the other hand it easily follows that (1.2) d(M, N ) ≤ 2d(M, N ) 1 + d(M, N ) ≤ 2d(M, N ). These concepts and some of their basic properties are considered, for example, in Gohberg and Krein [4] and Kato [8]. We will use the following properties of d(M, N ) that may be found in these references. Proposition 1.1. Let M and N be closed linear subspaces of a Banach space X. If dim M > dim N , then d(M, N ) = 1. Let M ⊥ denote the annihilator of M in the dual space X * , i.e., M ⊥ := {f : f ∈ X * , f(m) = 0 all m ∈ M }. Then from standard duality we have the following. Proposition 1.2. Let M and N be closed linear subspaces of a Banach space X. Then d(M, N ) = d(M ⊥ , N ⊥ ). The motivation in this paper is different from those of the above-mentioned references. Our goal is to estimate P M f − P N f in terms of d(M, N ) and f . Typically such an estimate will be of order d(M, N ) β with some β ≤ 1 which in general depends on the geometry of the space X. In this paper we mainly consider X = L p for p ∈ [1, ∞]. When X = H is a Hilbert space, this theory is well understood. A proof of the fact that P M − P N = d(M, N ) may be found in Akhiezer and Glazman [1]. (This text was originally published in Russian in 1950.) For completeness we present a short proof of this result and some extensions. Let X = H be a Hilbert space with inner product ( · , · ). Thus P M and P N are in fact linear operators and orthogonal projections. Proposition 1.3. In any Hilbert space we have (1.3) P M − P N = d(M, N ) for all closed linear subspaces M and N . Proof. Recall that by (1.1) we defined d(M, N ) as d(M, N ) = max ⎧ ⎨ ⎩ sup m∈M m =1 m − P N m , sup n∈N n =1 n − P M n ⎫ ⎬ ⎭ .

doi:10.1137/120873534
fatcat:f4rwioagrfbzffbff4yfvawqqe