Every finite dimensional vector space can made into an inner product space with the
same dimension.
[Math] True or False: Every finite dimensional vector space can made into an inner product space with the same dimension.
inner-productsvector-spaces
Best Answer
The definitions of inner product space that I have seen always require that vectors have a nonnegative inner product with themselves. Since inner products live in the base field, this requirement can only be meaningful if that field is of characteristic $0$. So your statement would be false over fields of prime characteristic. Also over fields larger than $\Bbb C$, like $\Bbb C(X)$, I believe it would be hard to arrange that the inner products of vectors with themselves lie in an ordered subfield like $\Bbb R$.
If (as is usual) you consider inner product spaces only over the fields $k=\Bbb R$ or $k=\Bbb C$, then indeed every finite dimensional vector space can be made into an inner product space, by transport via a vector space isomorphism with $k^n$ of the standard inner product on the latter.