Pythagorean theorem in functional analysisProve the Pythagorean theorem and its converse in R : f is orthogonal to g if and only if ‖ f − g ‖ 2 = ‖ f ‖ 2 + ‖ g ‖ 2 LHS -&gt; RHS ‖ f − g ‖ 2 = ( f , f ) − 2 ( f , g ) + ( g , g ) = ‖ f ‖ 2 − 2 ( f , g ) + ‖ g ‖ 2 In order for LHS=RHS −2(f,g) has to be 0 which means that f and g are othogonal. If −2(f,g)=0 Then = ‖ f ‖ + ‖ g ‖RHS-&gt; LHS ‖ f ‖ 2 + ‖ g ‖ 2 = ( f , f ) + ( g , g ) = ‖ f − g ‖ 2 iff − 2 ( f , g ) = 0 , iff f and g are orthogonal.

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Pythagorean theorem in functional analysisProve the Pythagorean theorem and its converse in R : f is orthogonal to g if and only if  ‖  f  −  g      ‖    2    =  ‖  f      ‖    2    +  ‖  g      ‖    2  LHS -&amp;gt; RHS                                  ‖        f        −        g                  ‖          2                                    =                                            (        f        ,        f        )        −        2        (        f        ,        g        )        +        (        g        ,        g        )                            =                                            ‖        f                  ‖          2                −        2        (        f        ,        g        )        +        ‖        g                  ‖          2                    In order for LHS=RHS −2(f,g) has to be 0 which means that f and g are othogonal. If −2(f,g)=0 Then  =  ‖  f  ‖  +  ‖  g  ‖RHS-&amp;gt; LHS                                  ‖        f                  ‖          2                +        ‖        g                  ‖          2                                    =                                            (        f        ,        f        )        +        (        g        ,        g        )                            =                                            ‖        f        −        g                  ‖          2                 iff         −        2        (        f        ,        g        )        =        0        ,         iff         f         and         g         are orthogonal.

le2ukogzbr · Accepted Answer

|        |    f  −  g      |              |        2    =  (  f  −  g  ,  f  −  g  )  =  (  f  ,  f  )  −  (  f  ,  g  )  −  (  g  ,  f  )  +  (  g  ,  g  )  =      |        |    f      |              |        2    +      |        |    g      |              |        2    −  2  (  f  ,  g  )So       |        |    f  −  g      |              |        2    =      |        |    f      |              |        2    +      |        |    g      |              |        2   f and g are orthogonal.

Pythagorean theorem in functional analysis Prove the Pythagorean theorem and its converse in R : f

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