The representations and continuity of the metric projections on two classes of half-spaces in Banach spaces.(English)Zbl 1474.46041

Summary: We show a necessary and sufficient condition for the existence of metric projection on a class of half-space $$K_{x_0^*, c} = \{x \in X : x^*(x) \leq c \}$$ in Banach space. Two representations of metric projections $$P_{K_{x_0^*, c}}$$ and $$P_{K_{x_0, c}}$$ are given, respectively, where $$K_{x_0, c}$$ stands for dual half-space of $$K_{x_0^*, c}$$ in dual space $$X^*$$. By these representations, a series of continuity results of the metric projections $$P_{K_{x_0^*, c}}$$ and $$P_{K_{x_0, c}}$$ are given. We also provide the characterization that a metric projection is a linear bounded operator.

MSC:

 46B20 Geometry and structure of normed linear spaces 41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
Full Text:

References:

 [1] Nevesenko, N. V., Continuity of the diameter of a metric projection, Matematicheskie Zapiski, 11, 4, 72–82, 131-132, (1979) · Zbl 0468.41022 [2] Oshman, E. V., Continuity of the metric projection, Matematicheskie Zametki, 37, 2, 200-2011, (1985) · Zbl 0586.41026 [3] Wang, J. H., Convergence theorems for best approximations in a nonreflexive Banach space, Journal of Approximation Theory, 93, 3, 480-490, (1998) · Zbl 0924.41021 [4] Fang, X. N.; Wang, J. H., Convexity and the continuity of metric projections, Mathematica Applicata, 14, 1, 47-51, (2001) · Zbl 1134.41338 [5] Zhang, Z.; Shi, Z., Convexities and approximative compactness and continuity of metric projection in Banach spaces, Journal of Approximation Theory, 161, 2, 802-812, (2009) · Zbl 1190.46018 [6] Wang, Y. W.; Yu, J. F., The character and representation of a class of metric projection in Banach space, Acta Mathematica Scientia A, Chinese Edition, 21, 1, 29-35, (2001) · Zbl 1018.46011 [7] Song, W.; Cao, Z. J., The generalized decomposition theorem in Banach spaces and its applications, Journal of Approximation Theory, 129, 2, 167-181, (2004) · Zbl 1067.46009 [8] Wang, J. H., The metric projections in nonreflexive Banach spaces, Acta Mathematica Scientia A, Chinese Edition, 26, 6, 840-846, (2006) · Zbl 1116.46301 [9] Ni, R. X., The representative of metric projection on the linear manifold in Banach spaces, Journal of Mathematical Research and Exposition, 25, 1, 99-103, (2005) [10] Cabrera, J.; Sadarangani, B., Weak near convexity and smoothness of Banach spaces, Archiv der Mathematik, 78, 2, 126-134, (2002) · Zbl 1021.46012 [11] Wang, Y. W.; Wang, H., Generalized orthogonal decomposition theorem in Banach space and generalized orthogonal complemented subspace, Acta Mathematica Sinica, Chinese Series, 44, 6, 1045-1050, (2001) · Zbl 1027.46013
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.