\HHT example; V= R x R h\\B#aaddcc

  \fvs3  \b Functionals from R x R --> R \  \_\_

  In the case V = R\]2 \ , each couple of numbers (c,d) from in R is a vector.
  Let us choose two basis vectors in V; for example e\[1 \ = (1,0) and e\[2 \ = (2,2).
  For every vector v =  a\[1 \ e\[1 \ +   a\[2 \ e\[2 \\_\_ 

  \vac
  f(v) = a\[1 \ f( e\[1 \ ) + a\[2 \ f( e\[2 \ )  \-\-\-	(*2) \_\_		
  v\
  
  It may be worth to see how this functionals look in "coordinate space". 
  Let us to fix basis { e\[1 \ , e\[2 \ ) temporarily and 
  consider "coordinate space" of couples (x,y) where x,y are coodinates in this basis.
  Let us use notation x=a\[1 \ and y=a\[2 \ for coodinates a and b\[1 \ and b\[2 \ for parameters
  b\[1 \ = f( e\[1 \ ) and b\[2 \ = f( e\[2 \ ) and display graph of function 
 
  f(x,y) = b\[1 \ x + b\[2 \ y in following applet.   \_\_

  For example in applet, coefficient b\[1 \ is chosen constant and coefficient b\[2 \ varies.\_
  Function f which is displayed red looks like a plain in space R\]3 \ . 

  \vac  \_\_

<applet codebase="viewer" code=linear_viewer.class width=600 height=600
    VSPACE="0" HSPACE="0" ALIGN = center>
<!width, height should fit  PARAM imageSize_x, imageSize_y>

 <param name="b1"        value=30>	   <! in percent units; functional f(x,y) slopes >
 <param name="b2"        value=80>

 <param name="b1Step"        value=200>	   <! in percent units; functionals slope steps in respect to frame/time number >
 <param name="b2Step"        value=0>

 <PARAM name="fps" 	       value="10">
 <PARAM name="FrameRingScale"  value="1">  <!FrameRingScale - graph shifts per frame; 0 - stops rotation >

 <PARAM name="Xmin"      value="0">    	   <!zeros taken here will restrict graph to a first quadrant >
 <PARAM name="Xmax"      value="600">
 <PARAM name="Ymin"      value="0">
 <PARAM name="Ymax"      value="600">
 <PARAM name="Zmin"      value="-270">     <!valid only for z-axis yet>
 <PARAM name="Zmax"      value="270">

 <PARAM name="Xstep"     value="10">
 <PARAM name="Ystep"     value="10">	   <!because first plain will be heavily covered by others; give it smallest step >

 <PARAM name="imageSize_x" value="600">
 <PARAM name="imageSize_y" value="600">
 
 <!delete this:>
 <PARAM name="web"	 value="survey.asp?tools=web.htm">
 <PARAM name="ftp"	 value="survey.asp?tools=ftp.htm">
 <PARAM name="browsers"  value="survey.asp?tools=browsers.htm">

</applet>
 
  v\\_\_


  Actually, graph of every function (*2) will be a "plain" in R\]3 \ . This can be proved different 
  ways. For example, if one will write equation (*2) in form \_\_

   \vac
   b\[1 \ x + b\[2 \ y  - f(x,y) = 0 \-\- or  \_\_

   uw = 0 \_\_

   v\

  where w is vector ( b\[1 \ ,  b\[2 \ ,  -1 ), and uw denotes a scalar product, then \_
  solution of this equations will be a set of vecors u = (x,y,f(x,y)) perpendicular
  vector w. All perpendicular vectors lie in one plain in R\]3 \ . \_\_

  According our general text, functionals f can be viewed as linear combinations of projection functionals. \_\_

  \vac 
  f = b\[1 \ P\[1 \ + b\[2 \ P\[2 \	\_\_
  v\
     
  In "coordinate space", "plain" P\[1 \ contains axis y and divides angle zOx in
  two equal parts.
  "Plain" P\[2 \ contains axis x and divides angle zOy equally. They look  "like" "blue" and "green"
  plaines in above applet correspondingly. 


h\