Document Zbl 1211.46010

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A, Theory Methods</a> <a href="/?q=in%3A266353" title="Articles in this Issue">73, No. 8, 2353-2360 (2010)</a>. </div> <div class="abstract">Let $\Phi: \mathbb R\to [0,\infty]$ be an Orlicz function, i.e., $\Phi$ is convex, even, left continuous on $[0,\infty)$, vanishes only at zero, and is not identically equal to infinity on $(0,\infty)$. Given a $\sigma$-finite non-atomic measure space $(G,\Sigma, \mu)$, let $L^0(\mu)$ denote the space of real valued $\mu$-measurable functions defined on $G$. Then the Orlicz space $L_\Phi= \{x \in L^0(\mu): \exists c>0\;I_\Phi(cx) = \int_G \Phi(cx(t)) \,d\mu(t) < \infty\}$ is a Banach space equipped with the standard Luxemburg or Orlicz norm.<br class="zbmathjax-paragraph">The dual space $L_\Phi^*$ is generated by the modular $\rho^*(f)= I_\Psi(v) + \|\varphi\|$, where $\Psi$ is the conjugate function to $\Phi$, and $f\in L_\Phi^*$ has a unique decomposition $f = v + \varphi$ with $v\in L_\Psi$ (which induces an integral functional) and $\varphi$ is a singular functional. Several useful relations among $I_\Phi$, $\rho^*$ and the Luxemburg and Orlicz norms are presented. They are applied to characterize the extreme points of the dual space $L_\Phi^*$ equipped with the Luxemburg norm. It is shown that a functional $f = v + \varphi \in L_\Phi^*$ such that $\|f\|=1$, $ 0\neq v \in L_\Psi, 0 \neq \varphi \in F$, where $F$ is the set of singular functionals, is an extreme point if and only if 7.5mm <dl class="zbmathjax-itemize"><dt class="zbmathjax-item1">(i)</dt><dd class="zbmathjax-item1"> $\rho^*(f) = 1$;</dd><dt class="zbmathjax-item1">(ii)</dt><dd class="zbmathjax-item1"> $\mu\{t\in G: v(t) \;\text{does not belong to the set of strict extreme points of}\;\Psi \} =0$;</dd><dt class="zbmathjax-item1">(iii)</dt><dd class="zbmathjax-item1"> $\varphi/\|\varphi\|$ is an extreme point of the unit ball of $L_\Phi^*$. </dd></dl><div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A11434">Anna Kamińska (Memphis)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A5774951">Cited in <strong>3</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A46B20" title="MSC2020">46B20</a> </td> <td class="space"> Geometry and structure of normed linear spaces </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A46B04" title="MSC2020">46B04</a> </td> <td class="space"> Isometric theory of Banach spaces </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A46E30" title="MSC2020">46E30</a> </td> <td class="space"> Spaces of measurable functions ($L^p$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A46A80" title="MSC2020">46A80</a> </td> <td class="space"> Modular spaces </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Aconvex+modular">convex modular</a>; <a href="/?q=ut%3AOrlicz+space%2C+dual+space">Orlicz space, dual space</a>; <a href="/?q=ut%3ALuxemburg+norm">Luxemburg norm</a>; <a href="/?q=ut%3AOrlicz+norm">Orlicz norm</a>; <a href="/?q=ut%3AAmemiya+formula">Amemiya formula</a>; <a href="/?q=ut%3Aextreme+point">extreme point</a>; <a href="/?q=ut%3Aregular+functional">regular functional</a>; <a href="/?q=ut%3Asingular+functional">singular functional</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1211.46010" data-ciurl="/ci/05774951" data-biburl="/bibtex/05774951.bib" data-amsurl="/amsrefs/05774951.bib" data-xmlurl="/xml/05774951.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/05774951.pdf" title="Zbl 1211.46010 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1016/j.na.2010.05.010" aria-label="DOI for “Some fundamental properties for duals of Orlicz spaces”" title="10.1016/j.na.2010.05.010">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">L. 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This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching. </div> </div></article> </div></div> </div> </div> <div class="clearfix"></div> </div> </div> <div id="foot"><div class="copyright"> © 2025 <a target="fiz" href="https://www.fiz-karlsruhe.de/en">FIZ Karlsruhe GmbH</a> <a href="/privacy-policy/">Privacy Policy</a> <a href="/legal-notices/">Legal Notices</a> <a href="/terms-conditions/">Terms & Conditions</a> <div class="info"> <ul class="nav"> <li class="mastodon"> <a href="https://mathstodon.xyz/@zbMATH" target="_blank" class="no-new-tab-icon"> <img src="/static/mastodon.png" title="zbMATH at Mathstodon (opens in new tab)" alt="Mastodon logo"> </a> </li> </ul> </div> </div> <div class="clearfix" style="height: 0px;"></div> </div> </div> <script src="https://static.zbmath.org/contrib/jquery/1.9.1/jquery.min.js"></script> <script src="https://static.zbmath.org/contrib/jquery-caret/1.5.2/jquery.caret.min.js"></script> <script src="/static/js/jquery-ui-1.10.1.custom.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap/v3.3.7zb1/js/bootstrap.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-lightbox/v0.7.0/bootstrap-lightbox.min.js"></script> <script src="https://static.zbmath.org/contrib/retina/unknown/retina.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-select/v1.13.14/js/bootstrap-select.min.js"></script> <script> var SCRIPT_ROOT = ""; </script> <script src="/static/scripts.js?v=20240926"> </script> <script src="https://static.zbmath.org/contrib/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script> <script type="text/x-mathjax-config"> MathJax.Hub.Config({ "HTML-CSS": { preferredFont: "TeX", availableFonts: [ "STIX", "TeX" ], linebreaks: { automatic: true }, EqnChunk: (MathJax.Hub.Browser.isMobile ? 10 : 50) }, tex2jax: { processEscapes: true, ignoreClass: "tex2jax_ignore|dno" }, TeX: { Macros: { Aut: "\\operatorname{Aut}", Hom: "\\operatorname{Hom}" }, noUndefined: { attributes: { mathcolor: "#039", //"red", mathbackground: "white", //"#FFEEEE", mathsize: "90%" } } }, messageStyle: "none" }); </script> <script type="text/javascript"> $(document).ready(function() { $("#MathInput").stop(true, true).keyup(function() { $.ajax({ url: "/mwsq/", type: "POST", data: { query : $("#MathInput").val() }, dataType: "text" }) .done(function(xml) { $("#MathPreview").html(xml); $(window).resize(); }); }); var press = jQuery.Event("keyup"); press.ctrlKey = false; press.which = 40; $("#MathInput").trigger(press); }); </script> <div id="new_tab_icon" style="display: none"> <span class="glyphicon glyphicon-new-window" aria-hidden="true"></span><span class="sr-only">(opens in new tab)</span></div> </body> </html>

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