on the entire space , since the whole space is dense in itself. The denseness of the domain is necessary and sufficient for the existence of the adjoint (if and are Hilbert spaces) and the transpose; see the sections below. If is closed, densely defined and continuous on its domain, then its domain is all of . A densely defined operator on a Hilbert space is called bounded from below if is a positive operator for some real number . That is, for all in the domain of (or alternatively since is arbitrary). If both and are bounded from below then is bounded. Example Let denote the space of continuous functions on the unit interval, and let denote the space of continuously differentiable functions. We equip with the supremum norm, , making it a Banach space. Define the classical differentiation operator by the usual formula: Every differentiable function is continuous, so . We claim that is a well-defined unbounded operator, with domain . For this, we need to show that is linear and then, for example, exhibit some such that and . This is a linear operator, since a linear combination of two continuously differentiable functions is also continuously differentiable, and The operator is not bounded. For example, satisfy but as . The operator is densely defined, and closed. The same operator can be treated as an operator for many choices of Banach space and not be bounded between any of them. At the same time, it can be bounded as an operator for other pairs of Banach spaces , and also as operator for some topological vector spaces . As an example let be an open interval and consider where: Adjoint The adjoint of an unbounded operator can be defined in two equivalent ways. Let be an unbounded operator between Hilbert spaces. First, it can be defined in a way analogous to how one defines the adjoint of a bounded operator. Namely, the adjoint of is defined as an operator with the property: More precisely, is defined in the following way. If is such that is a continuous linear functional on the domain of , then is declared to be an element of and after extending the linear functional to the whole space via the Hahn–Banach theorem, it is possible to find some in such that since Riesz representation theorem allows the continuous dual of the Hilbert space to