{x} =

It is easy to show from the Cauchy-Scwharz inequality for finite sequences that if the limit (|x|2) is a finite limit then also (|x|) exists as a finite limit. But it is not true that the existence of ( |x | 2 )asa finite limit implies that the limit (x) exists. A counterexample is given by the sequence x talcing values in {–1,1} with ever increasing strings of + 1s and – 1s so that the partial sums in (2.1) are not convergent. Now with y = 1, for which (1) and || 1 ||22 exist as limits, and x the ±1 sequence described above, then it is

clear that the partial sums

1 2M + 1

j=–M [xj + y j ] 2 = 12M + 1

do not converge. Hence the space of sequences for which || x ||22 exists as a limit is not a linear space.