On Simultaneous Farthest Points in ð ¿ âˆž ( ð ¼, ð ‘‹ )

Sh. Al-Sharif and M. Rawashdeh

International Journal of Mathematics and Mathematical Sciences, 2011, vol. 2011, 1-10

Abstract:

Let ð ‘‹ be a Banach space and let ð º be a closed bounded subset of ð ‘‹ . For ( ð ‘¥ 1 , ð ‘¥ 2 , … , ð ‘¥ ð ‘š ) ∈ ð ‘‹ ð ‘š , we set 𠜌 ( ð ‘¥ 1 , ð ‘¥ 2 , … , ð ‘¥ ð ‘š , ð º ) = s u p { m a x 1 ≤ ð ‘– ≤ ð ‘š ‖ ð ‘¥ ð ‘– − 𠑦 ‖ ∶ 𠑦 ∈ ð º } . The set ð º is called simultaneously remotal if, for any ( ð ‘¥ 1 , ð ‘¥ 2 , … , ð ‘¥ ð ‘š ) ∈ ð ‘‹ ð ‘š , there exists ð ‘” ∈ ð º such that 𠜌 ( ð ‘¥ 1 , ð ‘¥ 2 , … , ð ‘¥ ð ‘š , ð º ) = m a x 1 ≤ ð ‘– ≤ ð ‘š ‖ ð ‘¥ ð ‘– − ð ‘” ‖ . In this paper, we show that if ð º is separable simultaneously remotal in ð ‘‹ , then the set of ∞ -Bochner integrable functions, ð ¿ âˆž ( ð ¼ , ð º ) , is simultaneously remotal in ð ¿ âˆž ( ð ¼ , ð ‘‹ ) . Some other results are presented.

Date: 2011
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:890598

DOI: 10.1155/2011/890598

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