= Basis invariance of an isonormal Gaussian process
{title2=$W(h)=\sum_j Z_j\langle h,e_j\rangle$}
For any real separable <Hilbert space> $H$, any <orthonormal basis> $(e_j)$ and <independent> <standard normal random variables> $(Z_j)$, the displayed series converges <almost surely> for each fixed $h\in H$ and in <L2 space>. Indeed, the <Parseval identity for a Hilbertian basis> gives $\sum_j\langle h,e_j\rangle^2=\|h\|^2$, so the partial sums form an <L2 space>-bounded <martingale>. The <martingale convergence theorem> supplies both limits. Every finite linear combination has the <normal distribution> with <variance> $\|\sum_ju_jh_j\|^2$. Thus its finite-dimensional <characteristic function> is $\exp(-\|\sum_ju_jh_j\|^2/2)$, independent of the chosen <orthonormal basis>. This proves equality of the <finite-dimensional distributions> of the resulting <isonormal Gaussian processes>; pathwise equality needs a coupling.
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