The Journal of the Indian Mathematical Society

T. D. Narang ¹, Sahil Gupta ¹

Affiliations
1 Department of Mathematics, Guru Nanak Dev University, Amritsar-143 005, IN

Abstract

In this paper we discuss approximatively compact sets, strongly approximatively compact sets and strongly proximinal sets in metric linear spaces and prove some results on best approximation and continuity of the metric projection.

Keywords

Best Approximation, Proximinal Set, Strongly Proximinal Set, Chebyshev Set, Approximatively Compact Set, Strongly Approximatively Compact Set.

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T. D. Narang ¹

Affiliations
1 Department of Mathematics, Guru Nanak Dev University, Amritsar -143005, IN

Abstract

The paper deals with conditions under which an approximatively compact set G in a metric space (X, d) is Chebyshev. We also find invariant points of a self mapping on G which are also elements of best approximations in G.

Keywords

Best Approximation, Proximinal Set, Chebyshev Set, Metric Projection, Approximatively Compact Set, Approximatively Cauchy Set, Convex Set.

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T. D. Narang ¹, S. Chandok ²

Affiliations
1 Department of Mathematics, Guru Nanak Dev University, Amritsar-143005, IN
2 Department of Mathematics, Dr. B.R. Ambedkar National Institute of Technology, Jalandhar-144011, IN

Abstract

The paper contains some results on the continuity of the best approximation map in normed linear spaces, on fixed point of a continuous nowhere normal outward map in metric linear spaces, and on fixed point of a nonexpansive map in the case of metric spaces. The proved results generalize and extend the earlier known results on the subject.

Keywords

Core Point, Convex Metric Space, Best Approximation, Proximinal, Convex and Approximatively Compact Set.

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G. C. Ahuja ¹, T. D. Narang ¹, Swaran Trehan ¹

Affiliations
1 Department of Mathematics, University of Delhi, Delhi 110007, IN

Abstract

Section 2 of this paper deals with the generalizations of a theorem due to Motzkin, Straus and Valentine [7] and a theorem due to V. Klee [4]. Theorem 1 studies the continuity of the farthest point map in a matric space and Theorem 2 establishes that if a compact set in a Frechet space [9] is uniquely remotal with respect to the closure of its convex hull then the set is singleton.

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