One-sided James' compactness theorem

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• 作者：B. Cascales ; ^{beca@um.es" class="auth_mail" title="E-mail the corresponding author} ; J. Orihuela ^{joseori@um.es" class="auth_mail" title="E-mail the corresponding author} ; A. Pé ; rez ^{antonio.perez7@um.es" class="auth_mail" title="E-mail the corresponding author}
• 关键词：James' compactness theorem ; Weakly compact
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：15 January 2017
• 年：2017
• 卷：445
• 期：2
• 页码：1267-1283
• 全文大小：428 K

文摘

We present some extensions of classical results that involve elements of the dual of Banach spaces, such as Bishop–Phelp's theorem and James' compactness theorem, but restricting ourselves to sets of functionals determined by geometrical properties. The main result, which answers a question posed by F. Delbaen, is the following: Let E be a Banach space such that  (B_E^⁎,ω^⁎)$(B_{E^{⁎}},{\omega }^{⁎})$is convex block compact. Let A and B be bounded, closed and convex sets with distance  d(A,B)>0$d(A,B)>0$. If every  x^⁎∈E^⁎$x^{⁎}\in E^{⁎}$withiv class="formula" id="fm0010">iv class="mathml">sup⁡(x^⁎,B)<inf⁡(x^⁎,A)$\sup (x^{⁎},B)<\inf (x^{⁎},A)$iv>iv>attains its infimum on A and its supremum on B, then A and B are both weakly compact. We obtain new characterizations of weakly compact sets and reflexive spaces, as well as a result concerning a variational problem in dual Banach spaces.