When working with Fourier series, the inner product is defined as $$\int_{-L}^L f(x)g(x)dx$$
I see this definition everywhere and we know that $\rm{sin}\big(\frac{n\pi x}{L}\big)$ and $\rm{cos}\big(\frac{n\pi x}{L}\big)$ will form a orthogonal basis, but not orthonormal.
My question is: why is not more usual to define the inner product as $$\frac{1}{L}\int_{-L}^L f(x)g(x)dx$$ ?
Because with this definition, the previous basis will be orthonormal.
I think that the basis will be precisely $\Big\{ \frac{1}{2},\rm{sin}\big(\frac{\pi x}{L}\big),\rm{cos}\big(\frac{\pi x}{L}\big),\rm{sin}\big(\frac{2\pi x}{L}\big),\rm{cos}\big(\frac{2\pi x}{L}\big),\ldots \Big\} $.
An inner product is bilinear form that's symmetric definite positive, so whatever you multiply this inner product by positive real, it's still an inner product.