First Gaussian chaos ($\mathcal{G}={\overline{\{\ell (X):\ell \in B^{\ast }\}}}^{L^2}$)

For a centered Gaussian random variable in a Banach space, its first Gaussian chaos is the closed linear subspace of L2 space generated by its continuous linear observations. Every element is a centered Gaussian random variable, and every finite collection has a multivariate normal distribution: approximate in L2 space and pass to the characteristic function. The first Gaussian chaos supplies the scalar coordinates of its Cameron-Martin space of a Gaussian random variable in a Banach space.