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10.5 In Section 6.1 we defined the notion of an inner product on a real vector as
space by using the basic properties of the Euclidean inner product on
COMPLEX INNER axioms. In this section we shall define inner products on complex vector
PRODUCT SPACES spaces by using the properties of the Euclidean inner product on as
axioms.
Complex Inner Product Spaces
Motivated by Theorem 10.4.1, we make the following definition.
DEFINITION
An inner product on a complex vector space V is a function that associates a complex number with each pair of
vectors u and v in V in such a way that the following axioms are satisfied for all vectors u, v, and w in V and all scalars k.
(a)
(b)
(c)
(d) and if and only if
A complex vector space with an inner product is called a complex inner product space.
The following additional properties follow immediately from the four inner product axioms:
(i)
(ii)
(iii)
Since only (iii) differs from the corresponding results for real inner products, we will prove it and leave the other proofs as
exercises.
