Smooth functions on Banach spaces

This paper discusses the smoothness properties of partitions of unity which are available for any real separable Banach space B which is the support space for a mean zero Gaussian measure μ. Elements of the partition of unity are infinitely differentiable in the directions in which μ translates to an equivalent measure. The set of such directions forms a Hilbert subspace H of B, and the derivatives of the partition functions are shown to take values in the n-fold symmetric tensor product of H.