Motivation of my question: in my opinion, in view of the common definition, the statement "$\ell_p$ is a Hilbert space if and only if $p=2$" makes no sense because there is no inner product in the common definition of the $\ell_p$ space.
So, my question is: Is a Hilbert space (i) "an inner product space that is complete with respect to the norm induced by the inner product" or (ii) "a complete space with respect to a norm induced by some inner product"?
Notice that (i) is the common definition and (ii) is a definition for which the mentioned statement makes sense.
Thanks.
The point is that there is a canonical one-to-one correspondence between Hilbert spaces (i) and Hilbert spaces (ii), where (i), (ii) refer to the two definitions in your question.
Given a Hilbert space (i), if you take the norm derived from the inner product and “forget” the original inner product then you have a Hilbert space (ii).
If you have a Hilbert space (ii), you can determine the unique inner product inducing the norm by using the polarization identity, and this gives you a Hilbert space (i).
The above two constructions are each other's inverses, therefore nobody bothers to distinguish between the two “kinds” of Hilbert spaces.
(This answer is just an expanded version of Nate Eldredge's comment above.)