Gaussian random series on metric vector spaces (Q1085868)

Let \(u: H\to E\) be a continuous linear mapping of a separable Hilbert space into a separable metric vector space. Assume that E, endowed with its Borel \(\sigma\)-field, is a standard Borel space. Suppose that for an orthonormal basis \(\{x_ i\}\) of H the series \(\sum \xi _ iux_ i\) converges a.s., where \(\xi _ i\) is a sequence of independent standard Gaussian r.v.'s. We prove that then the series \(\sum \xi _ iuy_ i\) also converges a.s., for any orthonormal basis \(\{y_ i\}\) and both series have the same distributions.