According to wikipedia article on polarisation identity, in a normed space ( V , | | . | | , if the parallelogram law holds, then there is an inner product on V such that | | x | | 2 = ⟨ x , x ⟩ for all x ∈ V. However, from what I understand, we can prove polarisation identity this way: | | x + y | | 2 − | | x − y | | 2 = ⟨ x + y , x + y ⟩ − ⟨ x − y , x − y ⟩ = . . . = 4 ⟨ x , y ⟩However, from this proof, I don't see why we need parallelogram law. On top of that, shouldn't it be the other way round, such that if | | x | | 2 = ⟨ x , x ⟩, then the parallelogram law holds? Can someone explain to me please?

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According to wikipedia article on polarisation identity, in a normed space   (  V  ,      |        |    .      |        |  , if the parallelogram law holds, then there is an inner product on V such that       |        |    x      |              |        2    =  ⟨  x  ,  x  ⟩ for all   x  ∈  V. However, from what I understand, we can prove polarisation identity this way:      |        |    x  +  y      |              |        2    −      |        |    x  −  y      |              |        2    =  ⟨  x  +  y  ,  x  +  y  ⟩  −  ⟨  x  −  y  ,  x  −  y  ⟩  =  .  .  .  =  4  ⟨  x  ,  y  ⟩However, from this proof, I don&#039;t see why we need parallelogram law. On top of that, shouldn&#039;t it be the other way round, such that if       |        |    x      |              |        2    =  ⟨  x  ,  x  ⟩, then the parallelogram law holds? Can someone explain to me please?

Leroy Cunningham · Accepted Answer

The point of the theorem is the existence of the inner product. It is true (and easy to show) that if you are given an inner product   ⟨  −  ,  −  ⟩ then you can always define a norm via       |        |    x      |              |        2    =  ⟨  x  ,  x  ⟩, and this norm satisfies the parallelogram law and the polarisation identity. But that isn&#039;t the situation the theorem is considering. It is about the situation where you start with a norm and want to define an inner product such that   ⟨  x  ,  x  ⟩  =      |        |    x      |              |        2  .If you are just given a norm to start with, you can&#039;t necessarily define an inner product from that. Just defining   ⟨  x  ,  x  ⟩  =      |        |    x      |              |        2   doesn&#039;t even define   ⟨  x  ,  y  ⟩ in general, let alone mean that   ⟨  −  ,  −  ⟩ has the linearity properties you need for the &quot;=&quot; in your equation. Even defining   ⟨  x  ,  y  ⟩  =  (      |        |    x  +  y      |              |        2    −      |        |    x  −  y      |              |        2    )      /    4 doesn&#039;t get the linearity properties unless you have the parallelogram law as well.

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