be identified with the set of linear functionals given by the inner product. This vector is uniquely determined by if and only if the linear functional is densely defined; or equivalently, if is densely defined. Finally, letting completes the construction of which is necessarily a linear map. The adjoint exists if and only if is densely defined. By definition, the domain of consists of elements in such that is continuous on the domain of . Consequently, the domain of could be anything; it could be trivial (that is, contains only zero). It may happen that the domain of is a closed hyperplane and vanishes everywhere on the domain. Thus, boundedness of on its domain does not imply boundedness of . On the other hand, if is defined on the whole space then is bounded on its domain and therefore can be extended by continuity to a bounded operator on the whole space. If the domain of is dense, then it has its adjoint A closed densely defined operator is bounded if and only if is bounded. The other equivalent definition of the adjoint can be obtained by noticing a general fact. Define a linear operator as follows: Since is an isometric surjection, it is unitary. Hence: is the graph of some operator if and only if is densely defined. A simple calculation shows that this "some" satisfies: for every in the domain of . Thus is the adjoint of . It follows immediately from the above definition that the adjoint is closed. In particular, a self-adjoint operator (meaning ) is closed. An operator is closed and densely defined if and only if Some well-known properties for bounded operators generalize to closed densely defined operators. The kernel of a closed operator is closed. Moreover, the kernel of a closed densely defined operator coincides with the orthogonal complement of the range of the adjoint. That is, von Neumann's theorem states that and are self-adjoint, and that and both have bounded inverses. If has trivial kernel, has dense range (by the above identity.) Moreover: is surjective if and only if there is a such that for all in (This is essentially a variant of the so-called closed range theorem.) In particular, has closed range if and only if has closed range. In contrast to the bounded case, it is not necessary that since, for example, it is even possible that does not exist.