normal operators, but not to densely defined, closed operators in general, since in this case the spectrum can be empty. A symmetric operator defined everywhere is closed, therefore bounded, which is the Hellinger–Toeplitz theorem. Extension-related By definition, an operator T is an extension of an operator S if . An equivalent direct definition: for every x in the domain of S, x belongs to the domain of T and . Note that an everywhere defined extension exists for every operator, which is a purely algebraic fact explained at Discontinuous linear map#General existence theorem and based on the axiom of choice. If the given operator is not bounded then the extension is a discontinuous linear map. It is of little use since it cannot preserve important properties of the given operator (see below), and usually is highly non-unique. An operator T is called closable if it satisfies the following equivalent conditions: T has a closed extension; the closure of the graph of T is the graph of some operator; for every sequence (xn) of points from the domain of T such that xn → 0 and also Txn → y it holds that . Not all operators are closable. A closable operator T has the least closed extension called the closure of T. The closure of the graph of T is equal to the graph of Other, non-minimal closed extensions may exist. A densely defined operator T is closable if and only if T∗ is densely defined. In this case and If S is densely defined and T is an extension of S then S∗ is an extension of T∗. Every symmetric operator is closable. A symmetric operator is called maximal symmetric if it has no symmetric extensions, except for itself. Every self-adjoint operator is maximal symmetric. The converse is wrong. An operator is called essentially self-adjoint if its closure is self-adjoint. An operator is essentially self-adjoint if and only if it has one and only one self-adjoint extension. A symmetric operator may have more than one self-adjoint extension, and even a continuum of them. A densely defined, symmetric operator T is essentially self-adjoint if and only if both operators , have dense range. Let T be a densely defined operator. Denoting the relation "T is an extension of S" by S ⊂ T (a conventional abbreviation for Γ(S) ⊆ Γ(T)) one has the following. If T is symmetric then T