of the sequence of values of the functions at every point. If the functions take their values in a uniform space, then one can define pointwise Cauchy convergence, uniform convergence, and uniform Cauchy convergence of the sequence. Pointwise convergence implies pointwise Cauchy-convergence, and the converse holds if the space in which the functions take their values is complete. Uniform convergence implies pointwise convergence and uniform Cauchy convergence. Uniform Cauchy convergence and pointwise convergence of a subsequence imply uniform convergence of the sequence, and if the codomain is complete, then uniform Cauchy convergence implies uniform convergence. If the domain of the functions is a topological space, local uniform convergence (i.e. uniform convergence on a neighborhood of each point) and compact (uniform) convergence (i.e. uniform convergence on all compact subsets) may be defined. Note that "compact convergence" is always short for "compact uniform convergence," since "compact pointwise convergence" would mean the same thing as "pointwise convergence" (points are always compact). Uniform convergence implies both local uniform convergence and compact convergence, since both are local notions while uniform convergence is global. If X is locally compact (even in the weakest sense: every point has compact neighborhood), then local uniform convergence is equivalent to compact (uniform) convergence. Roughly speaking, this is because "local" and "compact" connote the same thing. Series of functions on a topological abelian group Pointwise and uniform convergence of series of functions are defined in terms of convergence of the sequence of partial sums. For functions taking values in a normed linear space, absolute convergence refers to convergence of the series of positive, real-valued functions . "Pointwise absolute convergence" is then simply pointwise convergence of . Normal convergence is convergence of the series of non-negative real numbers obtained by taking the uniform (i.e. "sup") norm of each function in the series (uniform convergence of ). 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. Normal convergence implies both local normal convergence and compact normal convergence. And if the domain is locally compact (even in the