An Introduction to Functional Analysis

Using the Riesz Representation Theorem, we define the Hilbert adjoint T^* of a linear map T from H into K (when H and K are both Hilbert spaces). This is another linear map from K into H. We show that the norm of T and its adjoint are equal. An operator is self-adjoint if it is equal to its adjoint (T=T^*); we compute explicitly the adjoints of some simple linear operators and give conditions under which they are self-adjoint.