Why do we define the inner product in the infinite-dimensional vector space $\displaystyle C([a,b], \mathbb{R}) $ as $\displaystyle \langle f,g \rangle = \int_{a}^{b} f(x)g(x) \ dx $?
it turns out that we do study inner products which are not symmetric. but we do not study arbitrary such products. for complex vector spaces we usually take conjugate symmetric inner products. i have not known of any other types as i guess it does not give us a rich enough structure to be useful