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We construct non-separable subspaces in the kernel of every quadratic functional on some classes of complex and real Banach spaces.
Suppose is a finite positive rotation invariant Borel measure on the open unit disc , and that the unit circle lies in the closed support of . For the Bergman space is the collection of functions in holomorphic on . We show that whenever a Gaussian power series almost surely lies in but not in , then almost surely: a) the zero set of is not contained in any zero set (, and b) is not contained in any zero set.
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