I am studying the Gram-Schmidt process of orthonormalizing a basis of a given vector space.
Suppose you have a vector space $W = \{ w_1,w_2...w_n\}$. Our goal is to find a set $U = \{u_1,u_2 ... u_n\}$ given by the formula
$ u_i = w_i - \sum_{j=1}^{i-1} \text{proj}_{u_j}(w_i) = w_i - \sum_{j=1}^{i-1}\frac{w_i\cdot u_j}{|u_j|^2}u_j$
I am sure all of you know this but note that the project part of the fomrula is just the inner space so we can rewrite as following
$ w_i - \sum_{j=1}^{i-1}\frac{\langle w_i,u_j\rangle}{\langle u_j \cdot u_j\rangle}u_j$ So my question is:
Can we have orthonormal basis of a vector space that is not a Euclidean space, or in other words how would you calculate the orthonormal basis for a polynomial vector space where the polynomials are in the form of, for example exponential functions.
I am mainly looking for examples so if you have any links to pdfs or if you know any examples your self please let me know.