A note on convergence of infinite series in a Banach space

It is discussed that the relations between the convergence, absolute convergence, weakly unconditional convergence and unconditional convergence and summability of an infinite series ∑∞n=1x_nin a Banach space X. The following conclusions are proved: (1) In a Banach space X, the unconditional convergence and summability of an infinite series are equivalent; (2) In a Hilbert space X, the weakly unconditional convergence, unconditional convergence and summability of an infinite series are equivalent; (3) In the case X is R or C, the unconditional convergence and absolute convergence as well as summability of an infinite series are equivalent.