A decomposition of the dual space of some Banach Function Spaces

We give a decomposition of the dual of some Banach Function Spaces, as the space of exponentially integrable functions, the Marcinkiewicz space and the Grand Lebesgue space, which are not reflexive. Furthermore, it is shown that the decomposition for the dual of the previous spaces holds in a more general setting, namely for any rearrangement invariant Banach Function Space such that its fundamental function in zero goes to zero.