Fernique's recent proof of finiteness of positive moments of the norm of a Banach-valued Gaussian random vector X is used to prove rth mean convergence of reproducing kernel series representationsof X. Embedding of the reproducing kernel Hilbert space into the Banach range of X is explicitly given by Bochner integration. This work extends and clarifies work of Kuelbs, Jain and Kallianpur. Fernique [2] has recently proved in a most elementary way that lima,<0+ E exp alIfX11%< oo for every centered Gaussian random vector X taking values in a real and separable Banach space B. As will be shown below, this result can be used to provide a dramatically simple proof of the strong convergence of certain representations of X by a series in B, as given by Kuelbs [4] and Jain-Kallianpur [3]. The role of reproducing kernel Hilbert spaces in such representations is sharply revealed by this approach. In this paper, B is a real and separable Banach space, B* its topological dual, X is the a-algebra generated by the open subsets of B, and P is a probability measure on a for which the induced distributions of the random variables x*CB* are all Gaussian with zero means. Suppose that in addition to being a Banach space, B is also a subset of the set of real functions on a set T (distinct points of B also being distinct as real functions on T), and that for each teT the evaluation mapping X, defined by XI(x)=x(t), xeB, is continuous on B. For example, if T is taken equal to B*, each xeB may be viewed as the continuous linear evaluation function on B* defined by x(x*)=x*(x), x*eB*. Let Y denote P quadratic-mean closure of {Xt, teT} viewed as a Hilbert subspace of Received by the editors February 6, 1971 and, in revised form, April 26, 1971. AMS 1970 classifications. Primary 60G15; Secondary 60G17.