In this paper, we state a maximal inequality for the partial sums of strongly mixing sequences of Hilbert space valued random variables. This inequality allows to derive the almost sure compactness of the partial sums divided by the normalizing sequence (n log log n)1/2. As a consequence, we derive the compact law of the iterated logarithm under the same condition than the one required in the real case, which is known to be essentially optimal. An application to Cramér-von Mises statistics is given.