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In mathematics — specifically, in functional analysis — an Asplund space or strong differentiability space is a type of well-behaved Banach space. Asplund spaces were introduced in 1968 by the mathematician Edgar Asplund, who was interested in the Fréchet differentiability properties of Lipschitz functions on Banach spaces. ==Equivalent definitions== There are many equivalent definitions of what it means for a Banach space ''X'' to be an Asplund space: * ''X'' is Asplund if, and only if, every separable subspace ''Y'' of ''X'' has separable continuous dual space ''Y''∗. * ''X'' is Asplund if, and only if, every continuous convex function on any open convex subset ''U'' of ''X'' is Fréchet differentiable at the points of a dense ''G''''δ''-subset of ''U''. * ''X'' is Asplund if, and only if, its dual space ''X''∗ has the Radon–Nikodým property. This property was established by Namioka & Phelps in 1975 and Stegall in 1978. * ''X'' is Asplund if, and only if, every non-empty bounded subset of its dual space ''X''∗ has weak-∗-slices of arbitrarily small diameter. * ''X'' is Asplund if and only if every non-empty weakly-∗ compact convex subset of the dual space ''X''∗ is the weakly-∗ closed convex hull of its weakly-∗ strongly exposed points. In 1975, Huff & Morris showed that this property is equivalent to the statement that every bounded, closed and convex subset of the dual space ''X''∗ is closed convex hull of its extreme points. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Asplund space」の詳細全文を読む スポンサード リンク
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