In this paper, the classical systems theoretic concepts of controllability and observability are considered for descriptor systems with an infinite-dimensional state space. These are systems of the form E(x) over dot(t) = Ax(t) + Bu(t), y(t) = Cx(t), where x (.), u (.), and y(.) are functions with values in separable Hilbert spaces X, U, and Y. For the operators, we assume that E : X -> Z, B : U -> Z, and C : X -> Y are bounded, where Z is another Hilbert space. The operator A is assumed to be closed and defined on some dense subspace D (A) subset of X. Mappings are defined that induce the notions of controllability and observability. The controllable states and the unobservable states are characterized by invariant subspaces. Based on that, Kalman decompositions of infinite-dimensional descriptor systems are presented.