LetXbe a real and infinite dimensional Banach space. ByλXwe denote the vector space of all sequences (αn) of real numbers such that (αnxn) lies inside the range of someX-valued measure with bounded variation for every null sequence (xn) inX. Among other results we prove: (i)λXis the largest normal sequence spaceμsatisfying that every sequence (xn) ∈μ{X} lies inside the range of someX**-valued measure with bounded variation, and (ii)λXis a perfect space for whichl1⊂λX⊂l2. We also determinate the sequence spaceλXwhenXis anLp?space (1 ≤p≤ +∞).