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<p>Let <i>K</i> be a (bounded) closed uniformly convex subset of a Banach space <i>X</i>. We show that<OrderedList> <ListItem> <ItemNumber>(i)</ItemNumber> <ItemContent> <p>the nearest point map is well-defined and always continuous from <i>X</i> onto <i>K</i>,</p> </ItemContent> </ListItem> <ListItem> <ItemNumber>(ii)</ItemNumber> <ItemContent> <p>there is a reflexive space <i>Y</i> with a uniform rotund in every direction norm such that <i>Y</i> contains <i>K</i> as a subset and the nearest point map <i>P</i><sub><i>K</i></sub>: <i>Y</i> → <i>K</i> is uniformly continuous from any bounded set containing <i>K</i> onto <i>K</i>.</p> </ItemContent> </ListItem> </OrderedList></p>

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Nearest Point Maps onto Uniformly Convex Sets

  • 期刊论文

  • 发表日期：

    2025年5月15日

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  • Qingjin Cheng,
  • Cuiling Wang,
  • Jianjian Wang

摘要

Let K be a (bounded) closed uniformly convex subset of a Banach space X. We show that (i)

the nearest point map is well-defined and always continuous from X onto K,

(ii)

there is a reflexive space Y with a uniform rotund in every direction norm such that Y contains K as a subset and the nearest point map PK: Y → K is uniformly continuous from any bounded set containing K onto K.