In mathematics, more specifically functional analysis and operator theory, the notion of unbounded operator provides an abstract framework for dealing with differential operators, unbounded observables in quantum mechanics, and other cases. The term "unbounded operator" can be misleading, since "unbounded" should sometimes be understood as "not necessarily bounded"; "operator" should be understood as "linear operator" (as in the case of "bounded operator"); the domain of the operator is a linear subspace, not necessarily the whole space; this linear subspace is not necessarily closed; often (but not always) it is assumed to be dense; in the special case of a bounded operator, still, the domain is usually assumed to be the whole space. In contrast to bounded operators, unbounded operators on a given space do not form an algebra, nor even a linear space, because each one is defined on its own domain. The term "operator" often means "bounded linear operator", but in the context of this article it means "unbounded operator", with the reservations made above. The given space is assumed to be a Hilbert space. Some generalizations to Banach spaces and more general topological vector spaces are possible. Short history The theory of unbounded operators developed in the late 1920s and early 1930s as part of developing a rigorous mathematical framework for quantum mechanics. The theory's development is due to John von Neumann and Marshall Stone. Von Neumann introduced using graphs to analyze unbounded operators in 1932. Definitions and basic properties Let be Banach spaces. An unbounded operator (or simply operator) is a linear map from a linear subspace —the domain of —to the space . Contrary to the usual convention, may not be defined on the whole space . An operator is said to be closed if its graph is a closed set. (Here, the graph is a linear subspace of the direct sum , defined as the set of all pairs , where runs over the domain of .) Explicitly, this means that for every sequence of points from the domain of such that and , it holds that belongs to the domain of and . The closedness can also be formulated in terms of the graph norm: an operator is closed if and only if its domain is a complete space with respect to the norm: An operator is said to be densely defined if its domain is dense in . This also includes operators defined