This is, however, the case if, for example, is bounded. A densely defined, closed operator is called normal if it satisfies the following equivalent conditions: ; the domain of is equal to the domain of and for every in this domain; there exist self-adjoint operators such that and for every in the domain of . Every self-adjoint operator is normal. Transpose Let be an operator between Banach spaces. Then the transpose (or dual) of is the linear operator satisfying: for all and Here, we used the notation: The necessary and sufficient condition for the transpose of to exist is that is densely defined (for essentially the same reason as to adjoints, as discussed above.) For any Hilbert space there is the anti-linear isomorphism: given by where Through this isomorphism, the transpose relates to the adjoint in the following way: where . (For the finite-dimensional case, this corresponds to the fact that the adjoint of a matrix is its conjugate transpose.) Note that this gives the definition of adjoint in terms of a transpose. Closed linear operators Closed linear operators are a class of linear operators on Banach spaces. They are more general than bounded operators, and therefore not necessarily continuous, but they still retain nice enough properties that one can define the spectrum and (with certain assumptions) functional calculus for such operators. Many important linear operators which fail to be bounded turn out to be closed, such as the derivative and a large class of differential operators. Let be two Banach spaces. A linear operator is closed if for every sequence in converging to in such that as one has and . Equivalently, is closed if its graph is closed in the direct sum . Given a linear operator , not necessarily closed, if the closure of its graph in happens to be the graph of some operator, that operator is called the closure of , and we say that is closable. Denote the closure of by . It follows that is the restriction of to . A core (or essential domain) of a closable operator is a subset of such that the closure of the restriction of to is . Example Consider the derivative operator where is the Banach space of all continuous functions on an interval . If one takes its domain to be , then is a closed operator which is not bounded. On the other hand