  We introduce a new notion of solution, which we call weak* solutions, for
systems of conservation laws. These solutions can be used to handle singular
situations that standard weak solutions cannot, such as vacuums in Lagrangian
gas dynamics or cavities in elasticity. Our framework allows us to treat the
systems as ODEs in Banach space. Starting with the observation that solutions
act linearly on test functions $\alpha\in X$, we require solutions to take
values in the dual space $X^*$ of $X$. Moreover, we weaken the usual
requirement of measurability of solutions. In order to do this, we develop the
calculus of the Gelfand integral, which is appropriate for weak* measurable
functions. We then use the Gelfand calculus to define weak* solutions, and show
that they are stronger than the usual notion of weak solution, although for
$BV$ solutions the notions are equivalent. It is expected that these solutions
will also shed light on vexing issues of ill-posedness for multi-dimensional
systems.
