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I am new to this topic. Generally, what is the process of proving two polynomials orthogonal to each other on some given interval? Feeding some input and see that inner products are zero can probably test the property but is not persuasive enough I guess. For example, if I am given: $$U(x)=4x^2-1$$ and $$w(x)=\sqrt{1-x^2}$$ on [-1, 1], how do I prove that the two are orthogonal (or not)?

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Inner product of these two functions means $\int_{-1}^1 U(x) \cdot w(x) \ dx$ (I believe it is called the standard inner product on the function space (consisting of continuous functions on [a,b], in this case [-1, 1]); a similar inner product is the one of the Hilbert-space), and orthogonality means the inner product is 0, as usual. See this link for more details.

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