Consider the following infinite-dimensional extension of the linear control system discussed in Hermes and La Salle [9].Let T be a set (time interval),$\mathcal{S}$ a $\sigma $-algebra of subsets of T, X a quasi-complete locally convex topological vector space, and ${\bf m} = (m_i )$ a sequence of vector measures $m_i :\mathcal{S} \to X$, $i = 1,2, \ldots $ For each $i = 1,2, \ldots $, a bounded real-valued $\mathcal{S}$-measurable function $f_i $ represents the effect of the i th control on the system. The total effect of all these controls is given by $\sum\nolimits_{i = 1}^\infty {\smallint _T f_i dm_i } $ the controls are restricted so that $(f_i (t)) \in \mathcal{F}(t)$, $\mathcal{F}(t)$ a subset of the product of countably many copies of the real line, for each $t \in T$, then the set of all values of the series above is the attainable set of this system, denoted by $A_\mathcal{F} ({\bf m})$ . The general bang-bang principle for this system is considered and conditions given for $A_\mathcal{F} ({\bf...