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<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E11">Th79</font></span>: <a NAME="T10"><span class="comment"><font color="firebrick">:: DUALSP03:10</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">X</font> being   <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a><br/>  for <font color="Olive" title="b2">x</font> being   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Olive" title="b1">X</font>  st  not <font color="Olive" title="b1">X</font> is  <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>  &amp; <font color="Olive" title="b2">x</font> is  <a href="dualsp03.html#V1" title="DUALSP03:attr.1">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">x</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#K3" title="DUALSP03:func.3">w-lim</a> <font color="Olive" title="b2">x</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">x</font></span><a href="normsp_0.html#K5" title="NORMSP_0:func.5">.||</a></span> &amp;  <a href="dualsp03.html#K3" title="DUALSP03:func.3">w-lim</a> <font color="Olive" title="b2">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="normsp_3.html#K19" title="NORMSP_3:func.19">ClNLin</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b2">x</font></span>)</span> )</div></div>
