this map is surjective (and thus bijective). Consequently, every antilinear functional on can be written (uniquely) in this form. If is the canonical linear bijective isometry that was defined above, then the following equality holds: Extending the bra–ket notation to bras and kets Let be a Hilbert space and as before, let Let which is a bijective antilinear isometry that satisfies Bras Given a vector let denote the continuous linear functional ; that is, so that this functional is defined by This map was denoted by earlier in this article. The assignment is just the isometric antilinear isomorphism which is why holds for all and all scalars The result of plugging some given into the functional is the scalar which may be denoted by Bra of a linear functional Given a continuous linear functional let denote the vector ; that is, The assignment is just the isometric antilinear isomorphism which is why holds for all and all scalars The defining condition of the vector is the technically correct but unsightly equality which is why the notation is used in place of With this notation, the defining condition becomes Kets For any given vector the notation is used to denote ; that is, The assignment is just the identity map which is why holds for all and all scalars The notation and is used in place of and respectively. As expected, and really is just the scalar Adjoints and transposes Let be a continuous linear operator between Hilbert spaces and As before, let and Denote by the usual bijective antilinear isometries that satisfy: Definition of the adjoint For every the scalar-valued map on defined by is a continuous linear functional on and so by the Riesz representation theorem, there exists a unique vector in denoted by such that or equivalently, such that The assignment thus induces a function called the of whose defining condition is The adjoint is necessarily a continuous (equivalently, a bounded) linear operator. If is finite dimensional with the standard inner product and if is the transformation matrix of with respect to the standard orthonormal basis then 's conjugate transpose is the transformation matrix of the adjoint Adjoints are transposes It is also possible to define the or of which is the map defined by sending a continuous linear functionals to where the composition is always a continuous linear functional on and it satisfies (this is true more