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<div><span class="kw">theorem </span><a NAME="T18"><span class="comment"><font color="firebrick">:: DUALSP03:18</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> holds <br/> ( <font color="Olive" title="b2">f</font> is  <a href="dualsp03.html#V2" title="DUALSP03:attr.2">weakly*-convergent</a>  iff ( <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;  ex <font color="Olive" title="b3">X0</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive" title="b1">X</font> st <br/>( <font color="Olive" title="b3">X0</font> is  <a href="normsp_3.html#V1" title="NORMSP_3:attr.1">dense</a>  &amp; (  for <font color="Olive" title="b4">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Olive" title="b1">X</font>  st <font color="Olive" title="b4">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b3">X0</font> holds <br/><font color="Olive" title="b2">f</font> <a href="dualsp03.html#K2" title="DUALSP03:func.2">#</a> <font color="Olive" title="b4">x</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  ) ) ) )</div></div>
