Help with Pythagorean theorem proof for Hilbert SpacesFor given functions f and g in a Hilbert space L 2 ( a , b ), prove if the Pythagorean theorem is true for f and g, then it is also true for cf and cg, where c is a constant. What would &lt; c f , c g &gt; be?Assume the Pythagorean Theorem holds for functions f and g in a Hilbert space L 2 ( a , b ). Then | | f | | 2 + | | g | | 2 = | | f + g | | 2 .I have the beginning of the proof above however I am having trouble continuing.

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Help with Pythagorean theorem proof for Hilbert SpacesFor given functions f and g in a Hilbert space       L    2    (  a  ,  b  ), prove if the Pythagorean theorem is true for f and g, then it is also true for cf and cg, where c is a constant. What would   &amp;lt;  c  f  ,  c  g  &amp;gt; be?Assume the Pythagorean Theorem holds for functions f and g in a Hilbert space       L    2    (  a  ,  b  ). Then       |        |    f      |              |        2    +      |        |    g      |              |        2    =      |        |    f  +  g      |              |        2  .I have the beginning of the proof above however I am having trouble continuing.

Leroy Lowery · Accepted Answer

The assumption is that  ‖  f      ‖    2    +  ‖  g      ‖    2    =  ‖  f  +  g      ‖    2    ..Now, since   ‖  c  f  ‖  =      |    c      |      ‖  f  ‖ for any scalar c and vector f,  ‖  c  f      ‖    2    +  ‖  c  g      ‖    2    =      |    c      |      ‖  f      ‖    2    +      |    c      |      ‖  g      ‖    2    =      |    c      |      (  ‖  f      ‖    2    +  ‖  g      ‖    2    )  =      |    c      |      ‖  f  +  g      ‖    2    =  ‖  c  f  +  c  g      ‖    2    ..

Help with Pythagorean theorem proof for Hilbert Spaces For given functions f and g in a Hilbert spa

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