Is the operation of addition on the subspaces of $V$ associative? In other words, if $U_1,U_2,U_3$ are subspaces of $V$, is

Yes, since either way we're taking all the sums (see def 1.3.6) the order which we do so doesn't matter.

Put another way, "all sums of (all sums of A,B) C = all sums of A,B,C".