A generalization of local uniform convexity of the norm

Panda, Biswanath; Kapoor, O. P.

doi:10.1016/0022-247x(75)90098-0

Cited by 24 publications

(12 citation statements)

References 6 publications

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“…Our next theorem shows that in an incomplete CLUR space, a virtual ball contains neither a farthest point (to any element of X), nor a nearest point (to any element of its complement). Recall that a normed linear space X is called CLUR (or X is said to have property (M) in the terminology of Panda and Kapoor (1975)); if x, x n e X, \\x\\ = 1, ||x n || < 1, and lim n ||x B + x\\ = 2 implies that {x n } has a convergent subsequence. The CLUR normed linear spaces were first introduced by L. P. Vlasov (1967), and were studied in some detail by Panda and Kapoor (1975).…”

Section: Resultsmentioning

confidence: 99%

“…Recall that a normed linear space X is called CLUR (or X is said to have property (M) in the terminology of Panda and Kapoor (1975)); if x, x n e X, \\x\\ = 1, ||x n || < 1, and lim n ||x B + x\\ = 2 implies that {x n } has a convergent subsequence. The CLUR normed linear spaces were first introduced by L. P. Vlasov (1967), and were studied in some detail by Panda and Kapoor (1975). In the proof of our Theorem 2 we shall make use of the following lemma.…”

Section: Resultsmentioning

confidence: 99%

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Completeness of normed linear spaces admitting centers

Astaneh

1985

J Aust Math Soc A

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It is shown that a normed linear space admitting (Chebyshev) centers is complete. Then the ideas in the proof of this fact are used to show that every incomplete CLUR (compactly locally uniformly rotund) normed linear space contains a closed bounded convex subset B with the following properties: (a) B does not contain any farthest point; (b) B does not contain any nearest point (to the elements of its complement).

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Section: Resultsmentioning

confidence: 99%

Section: Resultsmentioning

confidence: 99%

Completeness of normed linear spaces admitting centers

Astaneh

1985

J Aust Math Soc A

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“…Indeed it can be easily proved that X is locally uniformly convex if and only if it is strictly convex and has property (M) . The spaces with property (W) have been discussed in another paper by the authors [7]. Lemma 2 of the present paper contains one geometric property of the unit sphere of such spaces.…”

Section: Introductionmentioning

confidence: 87%

Approximative compactness and continuity of metric projections

Panda

Kapoor

1974

Bull. Austral. Math. Soc.

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In the paper "Some remarks on approximative compactness", Rev.Roumaine Math. Pyres Appl. 9 (196U), Ivan Singer proved that if K is an approximatively compact Chebyshev set in a metric space, then the metric projection onto K is continuous. The object of this paper is to show that though, in general, the continuity of the metric projection supported by a Chebyshev set does not imply that the set is approximatively compact, it is indeed so in a large class of Banach spaces, including the locally uniformly convex spaces. It is also proved that in such a space X the metric projection onto a Chebyshev set is continuous on a set dense in X .

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“…In 1936, J A Clarkson introduced the concept of uniform convexity (UR), and many started studying the kinds of properties related with it. A R Lovaglia first introduced the concept of locally uniform convexity, Panda and Kapoor [7] generalized the notion of local uniform convexity, and introduced the notion of compact local uniform convexity. These properties play an important role in some branches of mathematics.…”

Section: Introductionmentioning

confidence: 99%

“…A Banach space X is said to be locally uniformly convex if and only if every point of S(X) is a point of local uniform convexity (see [7]). …”

Section: Introductionmentioning

confidence: 99%