This paper explores the strong law of large numbers in the infinite dimensional setting. It is shown that under several classical conditions--such as the Kolmogorov condition--the strong law holds if and only if the weak law holds.
"Some Stability Results for Vector Values Random Variables." Ann. Probab. 7 (1) 75 - 84, February, 1979. https://doi.org/10.1214/aop/1176995149