In lecture we defined the Hermitian inner product between two
functions 
and 
to be given by
We also noted 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 
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 
follow from just these four basic
properties. You may then use 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.
( Note to math majors in the class: be aware that in the
mathematics literature the Hermitian inner product is usually defined in the
reverse way so that it is linear in its first argument. And, yes, I
am aware that a proper subset of these conditions suffices to define
the inner product.)
a)
b)
c)
d)
