In lecture we defined the Hermitian inner product between two functions f(q) and g(q) to be given by
Note that this operation is very analogous to the familiar vector dot product
Demonstrate that the following properties hold for the
Hermitian inner product and state the analogous
property for the vector dot product. In the statements below f,
g and h will be functions, whereas lower case Greek letters will be
constants, so that, for instance, 
means 
.
Mathematicians call any operation ;SPMlt;|;SPMgt; holding the four properties
listed below a Hermitian inner product. Because mathematicians take
great care in their definitions, most of the properties of the
operation we have defined for ;SPMlt;|;SPMgt; follow from just these four basic
properties. You may use this analogy as a guide for any proofs you may
need to give involving the Hermitian inner product. Because they are
so analogous to the properties of the ordinary vector inner product
(dot product), these relations should be easy to remember.
a)
b)
c)
d)
Note: c) above is redundant and not needed to define the
Hermitian Inner Product (HIP). I include it to remind you that pulling a
constant out of the left hand side of a HIP generates the complex
conjugate of the constant.