# Normed space, inner product space, which is larger?

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?

• 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. – BigbearZzz May 3 '16 at 8:06
• You can't just "restrict yourself to where $x=y$". You still have to define what $\langle x,y\rangle$ is for every pair $x,y$. – 5xum May 3 '16 at 8:07