The space L_1(L_p) is primary for 1<p<∞.

The classical Banach space L_1(L_p) consists of measurable scalar functions f on the unit square for which ‖f‖=∫^1_0(∫^1_0|f(x,y)|^pdy)^{1/p}dx<∞. We show that L_1(L_p)(1<p<∞) is primary, meaning that whenever L1(Lp)=E⊕F, where E and F are closed subspaces of L_1(L_p), then either E or F is isomorphic to L_1(L_p). More generally, we show that L_1(X) is primary for a large class of rearrangement-invariant Banach function spaces.