Adjoints

Let us study next the relation between the notions of linear transformation and dual space. Let be any vector space and let be any element of ; for any linear transformation on we consider the expression . For each fixed , the function defined by is a linear functional on ; using the square bracket notation for as well as for , we have . If now we allow to vary over , then this procedure makes correspond to each a , depending, of course, on ; we write . The defining property of is We assert that is a linear transformation on . Indeed, if , then The linear transformation is called the adjoint (or dual) of ; we dedicate this section and the next to studying properties of . Let us first get the formal algebraic rules out of the way; they go as follows.

Here (7) is to be interpreted in the following sense: if is invertible, then so is , and the equation is valid. The proofs of all these relations are elementary; to indicate the procedure, we carry out the computations for (6) and (7). To prove (6), merely observe that To prove (7), suppose that is invertible, so that . Applying (3) and (6) to these equations, we obtain Theorem 1 of Section: Inverses implies that is invertible and that (7) is valid.

In finite-dimensional spaces another important relation holds: This relation has to be read with a grain of salt. As it stands is a transformation not on but on the dual space of . If, however, we identify and according to the natural isomorphism, then acts on and (8) makes sense. In this interpretation the proof of (8) is trivial. Since is reflexive, we obtain every linear functional on by considering as a function of , with fixed in . Since defines a function (a linear functional) of , it may be written in the form . The vector here is, by definition, . Hence we have, for every in and for every in , the equality of the first and last terms of this chain proves (8).

Under the hypothesis of (8) (that is, finite-dimensionality), the asymmetry in the interpretation of (7) may be removed; we assert that in this case the invertibility of implies that of and, therefore, the validity of (7). Proof: apply the old interpretation of (7) to and in place of and .

Our discussion is summed up, in the reflexive finite-dimensional case, by the assertion that the mapping is one-to-one, and, in fact, an algebraic anti-isomorphism, from the set of all linear transformations on onto the set of all linear transformations on . (The prefix "anti" got attached because of the commutation rule (6).)