Adjoints and Riesz Representation Theorem

Riesz Representation Theorem

Linear Functional

A linear functional is a linear transformation from a Hilbert space to the underlying field of scalars.

Riesz Representation Theorem

Let be a continuous linear functional on a Hilbert space . Then, there exists a unique , such that for all . Moreover, .

Proof Consider the kernel of , say , which is a closed subspace of , so . If , then . Otherwise, pick some with , and let . Then if we let , observe that because , therefore, which shows that . Moreover, if is another element such that , then we have it follows that for all , which implies that .

Adjoint

Proposition

Let be a bounded linear map on separable Hilbert spaces. Then there exists a unique adjoint bounded linear map such that for all and .

Proof Given , the map is a linear functional on . So by the Riesz representation theorem, there exists a unique , such that for all . We define . This is a linear map from to , because the inner product is linear. It is also bounded because (see below).

Proposition

For any bounded linear operator , there holds , and

Proof We shall first show that . In fact, by the lemma Now, for any and , we have thus . For the other direction, note that so hence . The same argument applies to .

Infinite Diagonal Matrix

Diagonalised Linear Operator

Suppose is a separable Hilbert space with orthonormal basis . A linear operator is diagonalised if there exists a sequence of scalars such that

Proposition

Suppose is a diagonalised linear operator on a separable Hilbert space with orthonormal basis . Then the following holds:

• ,
• is also diagonalised with as the diagonal entries. Hence, if and only if for all .
• is unitary if and only if for all .
• is an orthogonal projection if and only if for all .