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<b>let </b><font color="Maroon">X</font> be   <a href="clvect_1.html#NM5">ComplexNormSpace</a>, <font color="Maroon">Y</font> be   <a href="clvect_1.html#NM5">ComplexNormSpace</a>;<br/><b>let </b><font color="Maroon">g</font> be  <a href="clopban1.html#V3" target="_self">bounded</a> <a href="clopban1.html#NM1">LinearOperator</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font>;<br/>



<a NAME="E1:56"/>
 <a href="clopban1.html#K12" target="_self">PreNorms</a> <font color="Maroon">g</font> is <a href="seq_4.html#V1">bounded_above</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">K</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E2:56_1"/><i><font color="Green">E38</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">K</font>
 <b>and </b><br/><a NAME="E3:56_1"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">x</font> being  <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font> holds  <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">K</font> <a href="real_1.html#K4">*</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Olive">x</font></span><a href="clvect_1.html#K5">.||</a></span>
 <b>by </b><i/>;<br/>
<div><i><font color="Green">E70</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:56_1_1"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="clopban1.html#K12" target="_self">PreNorms</a> <font color="Maroon">g</font>
 ;<br/><b>consider </b><font color="Maroon">t</font> being   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:56_1_1"/><i><font color="Green">E85</font></i>: 
( <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> &amp; <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">t</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i><a class="ref" href="clopban1.html#T1" target="_self">Th1</a></i>;<br/><a NAME="E4:56_1_1"/><i><font color="Green">E86</font></i>: 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">t</font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">K</font> <a href="real_1.html#K4">*</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">t</font></span><a href="clvect_1.html#K5">.||</a></span>
 <b>by </b><i><a class="txt" href="clopban1.html#E2:56_1"><i><font color="Green">E38</font></i></a></i>;<br/><a NAME="E5:56_1_1"/>
<font color="Maroon">K</font> <a href="real_1.html#K4">*</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">t</font></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">K</font> <a href="real_1.html#K4">*</a> 1
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><b>hence </b><a NAME="E6:56_1_1"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">K</font>
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div>
<b>take </b>
<font color="Maroon">K</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="seq_4.html#D1">SEQ_4:def 1</a><br/>

<b>thus </b><a NAME="E5:56_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>   holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="clopban1.html#K12" target="_self">PreNorms</a> <font color="Maroon">g</font> or <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">K</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:56"/>
( not  <a href="clopban1.html#K12" target="_self">PreNorms</a> <font color="Maroon">g</font> is <a href="xboole_0.html#V1">empty</a> &amp;  <a href="clopban1.html#K12" target="_self">PreNorms</a> <font color="Maroon">g</font> is <a href="seq_4.html#V1">bounded_above</a> )
 ;<br/>


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