A normalized positive linear functional – more concisely called a state – is the mathematical structure that quantum theory uses to assign probabilities to the various possible outcomes of a measurement, based on whatever we already know about the current status of the physical system. This article introduces the mathematical concept of a state and its relationship to state-vectors. For motivation, Gleason's theorem and its relatives are briefly reviewed. A general form of the Cauchy-Schwarz inequality is derived.
