# How the inner product changes under non-linear transformation

by mnb96
Tags: nonlinear, product, transformation
 P: 626 Hi, if we suppose x and y are two elements of some vector space V (say ℝn), and if we consider a linear function f:V→V', we know that the inner product of the transformed vectors is given by: $$\left\langle f\mathbf{x} , f\mathbf{y} \right\rangle = \left\langle \mathbf{x} , \overline{f}f\mathbf{y} \right\rangle = \left\langle \overline{f}f\mathbf{x} , \mathbf{y} \right\rangle$$ where $\overline{f}$ is the adjoint operator of $f$. What can we say about $\left\langle f\mathbf{x} , f\mathbf{y} \right\rangle$ when f is non-linear, for example a diffeomorphism ?