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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.

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    Yes, it's possible and sometimes pretty useful. The Gram-Schmidt process works just the same with any inner product. The orthonormal basis depends on which inner product you use for the vector space.2011-10-02

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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.

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    @Tyler: First, the Wikipage has $\frac{1}{2}(3x^2-1)$. Note that the Wikipedia page notes that the polynomials satisfy$\int_{-1}^1 P_nP_m = \frac{2}{2n+1}\delta_{nm}.$ So when you divide the polynomial in the Wikipage by its norm, you should get the polynomial you are getting (since you are getting the **normal** vector). Who said anything about "left hand side of the equation"? The polynomials in the Wiki page are not normalized (not divided by their norm); yours are.2011-10-02