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.
Best Answer
Yes, you can apply Gram-Schmidt to any inner product space. But there is no such thing as the orthonormal basis, just like there is no such thing as the basis of a vector space (except in the trivial, 0-dimensional case).
Moreover, you need to specify an inner product before you can have an orthonormal basis.
In the vector space of polynomials of degree at most $n$ with real or complex coefficients, for example, one possibility is to define $$\langle f,g\rangle = \int_{-1}^1 f(t)\overline{g(t)}\,dt$$ where $\overline{g(t)}$ is the complex conjugate. If you do this and start with the standard basis, you are led to the Legendre polynomials. Or you can define the inner product as $$\langle f,g\rangle = \int_0^1 f(t)\overline{g(t)}\,dx$$ which leads to different polynomials.
I'm not sure how to parse your question, though: how can the basis of a space of polynomials be given "in the form of exponential functions"? Exponential functions are not polynomials.
But given any basis and an inner product on a vector space, you can carry out Gram-Schmidt to any finite set of vectors and, after tossing away any zero vector you get, obtain an orthonormal basis for their span.