For any x, the series defined by x is dominated by the series m, and converges absolutely. So there is a limit function g, but does the series converge uniformly?
Given ε, find n so that the partial sum m1+m2+…+mn is within ε of its limit. Now fn+1(x)+fn+2(x)+fn+3(x)+… cannot sum to more than the tail of m, which is below ε. This doesn't depend on x, so the functions converge uniformly.
If the functions are finite vector functions, let mn found the norm of fn for all x in the domain. Now m is absolutely convergent, hence the norms of the sequence fn(x), for a fixed x, are absolutely convergent. By the norm equivalence principle, the sequence fn(x) is absolutely convergent, as it was above. Once again there is a limit function g.
Given ε, choose n as we did above. The tail, beyond fn(x), has norms that sum to no more than ε. The tail is absolutely convergent, and sums to something whose norm is less than ε. This does not depend on x, so f converges uniformly to g.