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<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E15">Th711</font></span>: <a NAME="T15"><span class="comment"><font color="firebrick">:: DUALSP03:15</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">X</font> being   <a href="lopban_1.html#NM2" title="LOPBAN_1:NM.2">RealBanachSpace</a><br/>  for <font color="Olive" title="b2">f</font> being   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><a href="dualsp01.html#K17" title="DUALSP01:func.17">DualSp</a> <font color="Olive" title="b1">X</font></span>)</span>  st <font color="Olive" title="b2">f</font> is  <a href="dualsp03.html#V2" title="DUALSP03:attr.2">weakly*-convergent</a>  holds <br/>( <span class="p1"><a href="normsp_0.html#K5" title="NORMSP_0:func.5">||.</a><span class="default"><font color="Olive" title="b2">f</font></span><a href="normsp_0.html#K5" title="NORMSP_0:func.5">.||</a></span> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a>  &amp; <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><a href="dualsp03.html#K4" title="DUALSP03:func.4">w*-lim</a> <font color="Olive" title="b2">f</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rinfsup1.html#K6" title="RINFSUP1:func.6">lim_inf</a> <span class="p1"><a href="normsp_0.html#K5" title="NORMSP_0:func.5">||.</a><span class="default"><font color="Olive" title="b2">f</font></span><a href="normsp_0.html#K5" title="NORMSP_0:func.5">.||</a></span> )</div></div>
