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?