Given an inner product space $(V,<>)$ where $V$ is a vector space. Denote the scalar field of $V$ as $F$, define an inner product as the following,
Given a basis $\bf{V}$, define the inner product matrix as the following,
$\bf{V}$ is an orthonormal basis if its inner product matrix with respect to the inner product $<>$ is an identity matrix.
The Gram-Schmidt process guarantees the existence of an orthonormal basis with respect to $<>$ for any inner product $<>$.
My question is, how about symmetric orthonormal basis (suppose V is a vector space over field $F$, or just look at vector space $\Bbb R^n$)? Can we always find an symmetric orthonormal basis for any inner product $<>$? Thank you!