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<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="csspace.html#NM1" title="CSSPACE:NM.1">ComplexUnitarySpace</a>;<br/><b>let </b><font color="Maroon" title="c2">seq</font> be   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c3">z</font> be   <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a>;<br/>





<b>assume </b><a NAME="E1:10"/>
<font color="Maroon" title="c2">seq</font> is <a href="clvect_3.html#V1" title="CLVECT_3:attr.1">summable</a>
 ;<br/>

<a NAME="E2:10"/><b>then </b><i><font color="Green" title="E5">A1</font></i>: 
 <a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font> is <a href="clvect_2.html#V1" title="CLVECT_2:attr.1">convergent</a>
 
<b>by </b><i><a class="ref" href="clvect_3.html#D2" target="_self" title="CLVECT_3:def.2">Def2</a></i>;<br/>
<a NAME="E3:10"/><b>then </b>
<font color="Maroon" title="c3">z</font> <a href="clvect_1.html#K7" title="CLVECT_1:func.7">*</a> <span class="p1">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span> is <a href="clvect_2.html#V1" title="CLVECT_2:attr.1">convergent</a>
 
<b>by </b><i><a class="ref" href="clvect_2.html#T5" title="CLVECT_2:th.5">CLVECT_2:5</a></i>;<br/>
<a NAME="E4:10"/><b>then </b>
 <a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="clvect_1.html#K7" title="CLVECT_1:func.7">*</a> <font color="Maroon" title="c2">seq</font></span>)</span> is <a href="clvect_2.html#V1" title="CLVECT_2:attr.1">convergent</a>
 
<b>by </b><i><a class="ref" href="clvect_3.html#T3" target="_self" title="CLVECT_3:th.3">Th3</a></i>;<br/>
<b>hence </b><a NAME="E5:10"/>
<font color="Maroon" title="c3">z</font> <a href="clvect_1.html#K7" title="CLVECT_1:func.7">*</a> <font color="Maroon" title="c2">seq</font> is <a href="clvect_3.html#V1" title="CLVECT_3:attr.1">summable</a>
 <b>by </b><i><a class="ref" href="clvect_3.html#D2" target="_self" title="CLVECT_3:def.2">Def2</a></i>;<br/>

<b>thus </b> <a href="clvect_3.html#K1" title="CLVECT_3:func.1">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="clvect_1.html#K7" title="CLVECT_1:func.7">*</a> <font color="Maroon" title="c2">seq</font></span>)</span> = 
 <a href="clvect_2.html#K1" title="CLVECT_2:func.1">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="clvect_1.html#K7" title="CLVECT_1:func.7">*</a> <span class="p2">(<span class="default"><a href="bhsp_4.html#K1" title="BHSP_4:func.1">Partial_Sums</a> <font color="Maroon" title="c2">seq</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="clvect_3.html#T3" target="_self" title="CLVECT_3:th.3">Th3</a></i>
<br/>.= 
<font color="Maroon" title="c3">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><a href="clvect_3.html#K1" title="CLVECT_3:func.1">Sum</a> <font color="Maroon" title="c2">seq</font></span>)</span>
<b>by </b><i><a class="txt" href="clvect_3.html#E2:10"><i><font color="Green" title="E5">A1</font></i></a>, <a class="ref" href="clvect_2.html#T15" title="CLVECT_2:th.15">CLVECT_2:15</a></i>
;<br/>


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