A note on Mackey topologies on Banach spaces

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• 作者：A.J. Guirao^a ; ¹ ; ^{anguisa2@mat.upv.es" class="auth_mail" title="E-mail the corresponding author} ; V. Montesinos^a ; ² ; ^{vmontesinos@mat.upv.es" class="auth_mail" title="E-mail the corresponding author} ; V. Zizler^b ; ^{vasekzizler@gmail.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Banach space ; Mackey topology ; Completeness ; Norming subspace ; Mazur space ; Weak compactness
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：1 January 2017
• 年：2017
• 卷：445
• 期：1
• 页码：944-952
• 全文大小：340 K

文摘

There is a maybe unexpected connection between three apparently unrelated notions concerning a given pan id="mmlsi1" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304395&_mathId=si1.gif&_user=111111111&_pii=S0022247X16304395&_rdoc=1&_issn=0022247X&md5=f167f9ff9552fcbebe3cd6c20dac3c1a" title="Click to view the MathML source">w^⁎pan>pan class="mathContainer hidden">pan class="mathCode">$w^{⁎}$pan>pan>pan>-dense subspace Y   of the dual pan id="mmlsi2" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304395&_mathId=si2.gif&_user=111111111&_pii=S0022247X16304395&_rdoc=1&_issn=0022247X&md5=65dfb549455cc136408e7f102bc5cc02" title="Click to view the MathML source">X^⁎pan>pan class="mathContainer hidden">pan class="mathCode">$X^{⁎}$pan>pan>pan> of a Banach space X: (i) The norming character of Y  , (ii) the fact that pan id="mmlsi3" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304395&_mathId=si3.gif&_user=111111111&_pii=S0022247X16304395&_rdoc=1&_issn=0022247X&md5=2dcc376a0e821abcfb58cd8a020b5fc2" title="Click to view the MathML source">(Y,w^⁎)pan>pan class="mathContainer hidden">pan class="mathCode">$(Y,w^{⁎})$pan>pan>pan> has the Mazur property, and (iii) the completeness of the Mackey topology pan id="mmlsi4" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304395&_mathId=si4.gif&_user=111111111&_pii=S0022247X16304395&_rdoc=1&_issn=0022247X&md5=0cd0cb8356afd808a1ccd28654210453" title="Click to view the MathML source">μ(X,Y)pan>pan class="mathContainer hidden">pan class="mathCode">$\mu (X,Y)$pan>pan>pan>, i.e., the topology on X   of the uniform convergence on the family of all absolutely convex pan id="mmlsi1" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304395&_mathId=si1.gif&_user=111111111&_pii=S0022247X16304395&_rdoc=1&_issn=0022247X&md5=f167f9ff9552fcbebe3cd6c20dac3c1a" title="Click to view the MathML source">w^⁎pan>pan class="mathContainer hidden">pan class="mathCode">$w^{⁎}$pan>pan>pan>-compact subsets of Y  . To clarify these connections is the purpose of this note. The starting point was a question raised by M. Kunze and W. Arendt and the answer provided by J. Bonet and B. Cascales. We fully characterize pan id="mmlsi4" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304395&_mathId=si4.gif&_user=111111111&_pii=S0022247X16304395&_rdoc=1&_issn=0022247X&md5=0cd0cb8356afd808a1ccd28654210453" title="Click to view the MathML source">μ(X,Y)pan>pan class="mathContainer hidden">pan class="mathCode">$\mu (X,Y)$pan>pan>pan>-completeness or its failure in the case of Banach spaces X   with a pan id="mmlsi1" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304395&_mathId=si1.gif&_user=111111111&_pii=S0022247X16304395&_rdoc=1&_issn=0022247X&md5=f167f9ff9552fcbebe3cd6c20dac3c1a" title="Click to view the MathML source">w^⁎pan>pan class="mathContainer hidden">pan class="mathCode">$w^{⁎}$pan>pan>pan>-angelic dual unit ball—in particular, separable Banach spaces or, more generally, weakly compactly generated ones—by using the norming or, alternatively, the Mazur character of Y. We characterize the class of spaces where the original Kunze–Arendt question has always a positive answer. Some other applications are also provided.