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My textbook says normed space is inner product space unless it satisfies some parallelogram identity.

In this case, we can conclude inner product space is a subset of normed space.

However, norm is something like $\|x\|:=\sqrt{\langle x,x\rangle}$.

If I somehow see this as an ordered pair $\{x,y\}$ (inner product), and we restrict ourselves to where $x=y$ (norm), with this concept in my mind, I conclude that normed space is a subset of inner product (because of set restrictions).

Which part went wrong?

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  • $\begingroup$ You must have misunderstood something. A normed space can be given an inner product compatible with the norm only if the parallelogram identity is satisfied. $\endgroup$ – BigbearZzz May 3 '16 at 8:06
  • $\begingroup$ You can't just "restrict yourself to where $x=y$". You still have to define what $\langle x,y\rangle$ is for every pair $x,y$. $\endgroup$ – 5xum May 3 '16 at 8:07

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