 The concept of normal convergence was first introduced by René Baire in 1908 in his book "Leçons sur les théories générales de l'analyse".Given a set "S" and functions formula_1 (or to any normed vector space), the series

is called normally convergent if the series of uniform norms of the terms of the series converges, i.e.,

Normal convergence implies, but should not be confused with, uniform absolute convergence, i.e.uniform convergence of the series of nonnegative functions formula_4.