We
present some extensions of classical results that involve elements of the dual of Banach s
paces, such as Bisho
p–Phel
p's theorem and
James' com
pactness theorem, but restricting ourselves to sets of functionals determined by geometrical
pro
perties. The main result, which answers a question
posed by F. Delbaen, is the following:
Let E be a Banach space such that pan id="mmlsi1" class="mathmlsrc">pan 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">(BEp>⁎p>,ωp>⁎p>)pan>pan class="mathContainer hidden">pan class="mathCode">pan>pan>pan>
is convex block compact. Let A and B be bounded, closed and convex sets with distance pan id="mmlsi15" class="mathmlsrc">pan 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)>0pan>pan class="mathContainer hidden">pan class="mathCode">pan>pan>pan>
. If every pan id="mmlsi16" class="mathmlsrc">pan 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">xp>⁎p>∈Ep>⁎p>pan>pan class="mathContainer hidden">pan class="mathCode">pan>pan>pan>
withattains its infimum on A and its supremum on B, then A and B are both weakly compact. We obtain new characterizations of weakly com
pact sets and reflexive s
paces, as well as a result concerning a variational
problem in dual Banach s
paces.