Is it true that:-an inner product satisfies the properties of a norm if and only if the norm satisfies the parallelogram equality-a norm can be induced by a metric if and only if the metric satisfies d ( x + a , y + a ) = d ( x , y ) and d ( a x , a y ) = a d ( x , y )or are the implications one way?

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Is it true that:-an inner product satisfies the properties of a norm if and only if the norm satisfies the parallelogram equality-a norm can be induced by a metric if and only if the metric satisfies   d  (  x  +  a  ,  y  +  a  )  =  d  (  x  ,  y  ) and   d  (  a  x  ,  a  y  )  =  a  d  (  x  ,  y  )or are the implications one way?

Bruno Hughes · Accepted Answer

The first statement is true both ways. Specifically, suppose   (  X  ,      |        |    ⋅      |        |    ) is a normed linear space. Then the norm       |        |    ⋅      |        |   is induced by an inner product iff the parallelogram law holds in   (  X  ,      |        |    ⋅      |        |    ).For the second statement, this is not true. Call the condition   d  (  x  ,  y  )  =  d  (  x  +  a  ,  y  +  a  ) translation invariance, and the condition   d  (  x  ,  y  )  =  d  (  a  x  ,  a  y  ) homogeneity. Consider the following variant of the characteristic function  χ  (  x  ,  y  )  =      {                            1          ,                          x          ≠          y          ,                                      0          ,                          otherwise.                        Homogeneity fails for   χ  ,     x  ≠  y. Indeed,  χ  (  x  ,  y  )  =  1  ≠  a  =  χ  (  a  x  ,  a  y  )  .

Is it true that: -an inner product satisfies the properties of a norm if and only if the norm s

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