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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16300208&_mathId=si1.gif&_user=111111111&_pii=S0022247X16300208&_rdoc=1&_issn=0022247X&md5=ee8c7a4ffe415acb7f3077f10de2dbdc" title="Click to view the MathML source">(BE⁎,ω⁎)class="mathContainer hidden">class="mathCode">is convex block compact. Let A and B be bounded, closed and convex sets with distance class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16300208&_mathId=si15.gif&_user=111111111&_pii=S0022247X16300208&_rdoc=1&_issn=0022247X&md5=b01aeed387f76fe714a14df5810ff0d1" title="Click to view the MathML source">d(A,B)>0class="mathContainer hidden">class="mathCode">. If every class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16300208&_mathId=si16.gif&_user=111111111&_pii=S0022247X16300208&_rdoc=1&_issn=0022247X&md5=f2442a007bbff6817168feacd805e3c3" title="Click to view the MathML source">x⁎∈E⁎class="mathContainer hidden">class="mathCode">withclass="formula" id="fm0010">
class="mathml">
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16300208&_mathId=si17.gif&_user=111111111&_pii=S0022247X16300208&_rdoc=1&_issn=0022247X&md5=fd4a1e0f884509e254d606dda2685844" title="Click to view the MathML source">sup(x⁎,B)<inf(x⁎,A)class="mathContainer hidden">class="mathCode">![]()
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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.