I've been looking into the fundamentals of quantum mechanics recently. In regular undergrad text one usually finds the definition that a state is an element in some Hilbert space $H$ and an observable is a bounded self-adjoint linear map from $H$ to the complex numbers.
A more accurate definition (although more involved mathematically), that actually manages to include mixed states (which the previous definition doesn't), is that observables are bounded self-adjoint elements of the algebra $A$ of linear maps from $H$ to the complex numbers, while states are defined as positive definite, normalized, linear maps from $A$ to the complex numbers.
What is surprising to me is that with this definition, $S$ (the set of all states) includes not only the elements of $H$ (via an isomorphism), but also some extra states (which is easily seen with particular examples). The confusing part is that from these definitions it seems to me that $S$ is a subset of $(H^*)^*$, which should be isomorphic to a subset in $H$ (or at lest so I know from my linear algebra courses); however, the matter is reversed: $H$ is isomorphic to a subset of $S$. What am I missing here?