Okay, I think I get it… I thought one would/could define orthogonality of polynomials in terms of an inner product equivalent to the familiar K^n dot product, ie. sum of pairwise products of equal-degree coefficients, but I guess polynomials just don’t work that way. I can’t say I understand the hows and whys of the "correct" inner product given by Wikipedia :/
> I thought one would/could define orthogonality of polynomials in terms of an inner product equivalent to the familiar K^n dot product, ie. sum of pairwise products of equal-degree coefficients
Well, you can. But then you essentially get the real coordinate space, and them being polynomials and not just real-valued vectors is just an extra story that you slapped on top, but that has no relevance.
Polynomial vector spaces become useful, when we treat polynomials as functions, and the vector space of polynomials as a subspace of some other space of functions. And with function spaces the inner product is defined as an integral of the product of two functions (possibly with a weight function thrown in).
Or maybe the coefficient-based inner product spaces of polynomials have some uses, too. But they are less common than the function-based (integral) inner product.
