 In Banach spaces, pointwise absolute convergence implies pointwise convergence, and normal convergence implies uniform convergence.For functions defined on a topological space, one can define (as above) local uniform convergence and compact (uniform) convergence in terms of the partial sums of the series.If, in addition, the functions take values in a normed linear space, then local normal convergence (local, uniform, absolute convergence) and compact normal convergence (absolute convergence on compact sets) can be defined.