generally, when and are merely normed spaces). So for example, if then sends the continuous linear functional (defined on by ) to the continuous linear functional (defined on by ); using bra-ket notation, this can be written as where the juxtaposition of with on the right hand side denotes function composition: The adjoint is actually just to the transpose when the Riesz representation theorem is used to identify with and with Explicitly, the relationship between the adjoint and transpose is: which can be rewritten as: Alternatively, the value of the left and right hand sides of () at any given can be rewritten in terms of the inner products as: so that holds if and only if holds; but the equality on the right holds by definition of The defining condition of can also be written if bra-ket notation is used. Descriptions of self-adjoint, normal, and unitary operators Assume and let Let be a continuous (that is, bounded) linear operator. Whether or not is self-adjoint, normal, or unitary depends entirely on whether or not satisfies certain defining conditions related to its adjoint, which was shown by () to essentially be just the transpose Because the transpose of is a map between continuous linear functionals, these defining conditions can consequently be re-expressed entirely in terms of linear functionals, as the remainder of subsection will now describe in detail. The linear functionals that are involved are the simplest possible continuous linear functionals on that can be defined entirely in terms of the inner product on and some given vector Specifically, these are and where Self-adjoint operators A continuous linear operator is called self-adjoint it is equal to its own adjoint; that is, if Using (), this happens if and only if: where this equality can be rewritten in the following two equivalent forms: Unraveling notation and definitions produces the following characterization of self-adjoint operators in terms of the aforementioned continuous linear functionals: is self-adjoint if and only if for all the linear functional is equal to the linear functional ; that is, if and only if where if bra-ket notation is used, this is Normal operators A continuous linear operator is called normal if which happens if and only if for all Using () and unraveling notation and definitions produces the following characterization of normal operators in terms of inner products of continuous linear functionals: is a normal operator if and only