I am running into a very interesting phenomenon that I do not quite understand
(Illustration of an example of so called subset of $\mathbb{R}^n$)
For example, suppose we have a subset of $X \subseteq \mathbb{R}^n$ equipped with the usual inner product
$\langle x, y \rangle = \sum\limits_{i = 1}^n x_iy_i, x,y \in X$
The subset can alternatively be equipped with another inner product, say a weighted inner product that is defined on each point $z \in X$
$\langle x, y \rangle_z = \sum\limits_{i = 1}^n \dfrac{x_iy_i}{z_i}, x, y, z, \in X$
Does it make sense to define a function that is:
$f(x,z) = x^Tx + \sum\limits_{i = 1}^n \dfrac{x_ix_i}{z_i} = \langle x, x \rangle + \langle x, x \rangle _z$?
The reason I am asking is because I have such a function with the above structure defined on a space where multiple inner products are well defined. One is the usual, another is the weighted. I wonder if I could express $f$ using two inner products.