Martingale convergence, series expansion of Gaussian elements, and strong law of large numbers in Fréchet spaces

The main result of this paper is the derivation of a convergence theorem for certain martingales with values in a separable Fréchet space F. It is shown that this result includes a well known theorem due to Chatterji. Moreover, the series expansion of zero-mean Gaussian elements with values in F and the strong law of large numbers for i.i.d. F-valued random elements also follow as applications of the main theorem.