<?xml version="1.0"?>
<div class="add">

<b>set </b><font color="Maroon">g</font> =  <a href="bilinear.html#K11">FormFunctional</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="hermitan.html#K1" target="_self">*'</a> </span>)</span>;<br/>
<b>thus </b><a NAME="E1:74_1_1"/>
 <a href="bilinear.html#K11">FormFunctional</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="hermitan.html#K1" target="_self">*'</a> </span>)</span> is <a href="hermitan.html#V5" target="_self">diagReR+0valued</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">v</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="hermitan.html#D7" target="_self">HERMITAN:def 7</a><br/>

<a NAME="E1:74_1_1_1"/><i><font color="Green">E30</font></i>: 
<span class="p1">(<span class="default"><a href="bilinear.html#K11">FormFunctional</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="hermitan.html#K1" target="_self">*'</a> </span>)</span></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">v</font>,<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="complfld.html#K3">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">0 <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E2:74_1_1_1"/>
<span class="p1">(<span class="default"><a href="bilinear.html#K11">FormFunctional</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="hermitan.html#K1" target="_self">*'</a> </span>)</span></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">v</font>,<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bilinear.html#K11">FormFunctional</a> <font color="Maroon">f</font>,<span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="hermitan.html#K1" target="_self">*'</a> </span>)</span></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">v</font>,<font color="Maroon">v</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bilinear.html#K11">FormFunctional</a> <font color="Maroon">f</font>,<span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="hermitan.html#K1" target="_self">*'</a> </span>)</span></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">v</font>,<font color="Maroon">v</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="complex1.html#T29">COMPLEX1:29</a></i>;<br/>
<a NAME="E3:74_1_1_1"/><b>then </b>
 <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K11">FormFunctional</a> <font color="Maroon">f</font>,<span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="hermitan.html#K1" target="_self">*'</a> </span>)</span></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">v</font>,<font color="Maroon">v</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="square_1.html#K5">^2</a> 
 
<b>by </b><i><a class="ref" href="hermitan.html#T4" target="_self">Th4</a>, <a class="ref" href="complex1.html#T163">COMPLEX1:163</a></i>;<br/>
<b>hence </b><a NAME="E4:74_1_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K11">FormFunctional</a> <font color="Maroon">f</font>,<span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="hermitan.html#K1" target="_self">*'</a> </span>)</span></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">v</font>,<font color="Maroon">v</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T65">XREAL_1:65</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:74_1_1"/>
 <a href="bilinear.html#K11">FormFunctional</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="hermitan.html#K1" target="_self">*'</a> </span>)</span> is <a href="hermitan.html#V3" target="_self">hermitan</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">v</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font>, <font color="Maroon">w</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="hermitan.html#D5" target="_self">HERMITAN:def 5</a><br/>

<b>thus </b><span class="p1">(<span class="default"><a href="bilinear.html#K11">FormFunctional</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="hermitan.html#K1" target="_self">*'</a> </span>)</span></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">v</font>,<font color="Maroon">w</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="hermitan.html#K1" target="_self">*'</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="complfld.html#K2">*'</a> </span>)</span> <a href="complfld.html#K2">*'</a> 
<b>by </b><i><a class="ref" href="bilinear.html#D11">BILINEAR:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font></span>)</span> <a href="complfld.html#K2">*'</a> </span>)</span></span>)</span> <a href="complfld.html#K2">*'</a> </span>)</span> <a href="complfld.html#K2">*'</a> 
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span> <a href="complfld.html#K2">*'</a> </span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font></span>)</span> <a href="complfld.html#K2">*'</a> </span>)</span> <a href="complfld.html#K2">*'</a> </span>)</span></span>)</span> <a href="complfld.html#K2">*'</a> 
<b>by </b><i><a class="ref" href="complfld.html#T90">COMPLFLD:90</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="hermitan.html#K1" target="_self">*'</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span></span>)</span> <a href="complfld.html#K2">*'</a> 
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K11">FormFunctional</a> <font color="Maroon">f</font>,<span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="hermitan.html#K1" target="_self">*'</a> </span>)</span></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">w</font>,<font color="Maroon">v</font></span>)</span> <a href="complfld.html#K2">*'</a> 
<b>by </b><i><a class="ref" href="bilinear.html#D11">BILINEAR:def 11</a></i>
;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">v</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="hermitan.html#D6" target="_self">HERMITAN:def 6</a><br/>

<a NAME="E3:74_1_1"/><i><font color="Green">E31</font></i>: 
<span class="p1">(<span class="default"><a href="bilinear.html#K11">FormFunctional</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="hermitan.html#K1" target="_self">*'</a> </span>)</span></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">v</font>,<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="hahnban1.html#K1">[**</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="complfld.html#K3">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span>,0</span><a href="hahnban1.html#K1">**]</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E4:74_1_1"/>
<span class="p1">(<span class="default"><a href="bilinear.html#K11">FormFunctional</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="hermitan.html#K1" target="_self">*'</a> </span>)</span></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">v</font>,<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bilinear.html#K11">FormFunctional</a> <font color="Maroon">f</font>,<span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="hermitan.html#K1" target="_self">*'</a> </span>)</span></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">v</font>,<font color="Maroon">v</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bilinear.html#K11">FormFunctional</a> <font color="Maroon">f</font>,<span class="p5">(<span class="default"><font color="Maroon">f</font> <a href="hermitan.html#K1" target="_self">*'</a> </span>)</span></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">v</font>,<font color="Maroon">v</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="complex1.html#T29">COMPLEX1:29</a></i>;<br/>
<b>hence </b><a NAME="E5:74_1_1"/>
 <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K11">FormFunctional</a> <font color="Maroon">f</font>,<span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="hermitan.html#K1" target="_self">*'</a> </span>)</span></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">v</font>,<font color="Maroon">v</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="hermitan.html#T4" target="_self">Th4</a>, <a class="ref" href="complex1.html#T163">COMPLEX1:163</a></i>;<br/>


</div>
