Is a subset of an inner product space also an inner product space? My question may seem trivial but it's important that I know this. I know for a fact that a subspace of an inner product space is also an inner product space, but how about an arbitrary subset? Could I argue that since we are allowed to pick any subset, we could pick one that is not a subspace and since it is not a subspace it is not eligible as an inner product space?
Thank you!

A subset of an inner product space does not have to be an inner product space.
As you said, we can pick a subset that is not a subspace, hence not an inner product space.
