Convexities and approximative compactness and continuity of metric projection in Banach spaces

Abstract

In this paper, we investigate the continuities of the metric projection in a nonreflexive Banach space X , which improve the results in [X.N. Fang, J.H. Wang, Convexity and continuity of metric projection, Math. Appl. 14 (1) (2001) 47-51; P.D. Liu, Y.L. Hou, A convergence theorem of martingales in the limit, Northeast. Math. J. 6 (2) (1990) 227-234; H.J. Wang, Some results on the continuity of metric projections, Math. Appl. 8 (1) (1995) [80][81][82][83][84]. Under the assumption that X has some convexities, we discuss the relationship between approximative compactness of a subset A of X and continuity of the metric projection P A . We also give a representation theorem for the metric projection to a hyperplane in dual space X * and discuss its continuity.

The Representation and Continuity of a Generalized Metric Projection onto a Closed Hyperplane in Banach Spaces

“…In Section 3, under , we give a distance formula from a point in to a hyperplane ( * , ) in and a representation of the generalized metric projection ( * , ) and consider the continuity of ( * , ) . Results obtained in the present paper extend classical Ascoli Theorem (i.e., the distance formula under the case of norm from a point to a closed hyperplane in a Banach space) and main results in [8,[10][11][12] from the setting of norm to that of the Minkowski functional.…”

“…We then show that −1 ( * ) is weakly compact. Since −1 ( * / ∘ ( * )) ⊆ bd by (14) and since −1 ( * / ∘ ( * )) = (1/ ∘ ( * )) −1 ( * ) by (12), one sees that −1 ( * / ∘ ( * )) is a convex subset of bd . It follows that −1 ( * / ∘ ( * )) is relatively weakly compact because is weakly nearly strictly convex; hence, −1 ( * ) is relatively weakly compact.…”

“…In this direction, some meaning results, such as existences, characterizations, and well-posedness of this kind of approximation, have been established recently; see [1,[5][6][7]. In this paper, we will consider the problem of representation of generalized metric projection few authors; see [8][9][10][11][12]. In particular, when is reflexive, Wang and Yu have given in [10] the representation of ( ) ( * , ) , which was further extended by Ni in [8] to the case of nonreflexive Banach spaces.…”

“…In particular, when is reflexive, Wang and Yu have given in [10] the representation of ( ) ( * , ) , which was further extended by Ni in [8] to the case of nonreflexive Banach spaces. When is nearly strictly convex, Wang has shown in [11] that ( ) ( * , ) is norm-to-weak upper semicontinuous on , while, when is arbitrary Banach space, Zhang and Shi have given in [12] the pointwise continuity of ( ) ( * , ) under an additional condition. It should be noted that, when one uses a nonnegative convex function on the Euclidean space R satisfying (0) = 0 and ( ) = ( ) for all ∈ R and ≥ 0 as a metric on R (i.e., the distance from a point to a subset of R is defined as ( , ) = inf ∈ ( − )), Ferreia and Nemeth have investigated in [13] the problem of the best approximation in R and, in particular, given some properties of corresponding metric projections on a hyperplane in R .…”

Geometric properties and continuity of the pre-duality mapping in Banach space

“…They have been studied, e.g., in [1,7,16,18,19,23]. In particular, [7,19,20,22] contain applications to approximation theory.…”