Let B be a real separable Banach space and R: i'->ia covariance operator. All representations of R in the form 2en ® e", (e", n > 1} c fi, are characterized. Necessary and sufficient conditions for R to be compact are ob- tained, including a generalization of Mercer's theorem. An application to character- istic functions is given. 1. Introduction. The study of covariance operators is a major component in the theory of probability measures on Banach spaces (10), (9), (1). The covariance operator of a strong second-order measure is always compact (2); however, the covariance operator of a weak second-order measure need not be compact. In this paper we first characterize series representations of covariance operators, and then give a set of necessary and sufficient conditions for a covariance operator to be compact. The classical Mercer's theorem (7) can be obtained as an immediate corollary. These results are then applied to extend a result of Prohorov and Sazanov (6) on relative compactness of probability measures from Hubert space to Banach space. 2. Definitions and notation. B is a real separable Banach space with norm || ■ || and topological dual B*. A linear operator R: B* -» B is a covariance operator if 7? is symmetric and nonnegative: {Ru, u> = and 0, for all u, v in B*. A probability measure ii on the Borel a-field of B is said to be weak second-order if fB(x, «>2 dii(x) > = j = I {x — m, u}(x — m, v} d(i(x),