I'm wondering if the integral of the product of two real-valued functions on a given space is also necessarily also an inner product for every function that can be defined on that space. E.g., are the Riemann integral, Lebesgue integral, Darboux integral, Riemann-Stieltjes integral, Daniell integral, Haar integral, Henstock-Kurzwell integral, Ito integral, and Young integral (see this article) all inner products on their respective spaces?
**I'm thinking about inner products as more general versions of integrals of products, but maybe I'm thinking about this the wrong way. Is it better to think about integrals of products as more general versions of inner products?