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defined on the open disc unit of radius 1 in the complex plane

take e_n=z^n and then normalize it, I'm trying to figure out if this is a basis for the

space. my intuition is no, but that would require showing for e.g. that theres a nontrivial

vector thats perpendicular to the entire space

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The normalized monomials do form an o.n. basis for the Bergman space: see the comments here for instance mathoverflow.net/questions/68172/… (To see that the perp of the set of monomials is zero, take a function in Bergman space and look at its Taylor series.) If you need more details then I suggest you ask on math.stackexchange.com where this question is more appropriate –  Yemon Choi Jan 15 '12 at 4:00

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