Page 112 - Linear Algebra Done Right

Inner Products
by abstracting the properties of the dot product discussed in the para-
graph above. For real vector spaces, that guess is correct. However,
so that we can make a definition that will be useful for both real and 99
complex vector spaces, we need to examine the complex case before
making the definition.
Recall that if λ = a + bi, where a, b ∈ R, then the absolute value
of λ is defined by

2
2
|λ|= a + b ,
the complex conjugate of λ is defined by
¯
λ = a − bi,
and the equation
¯
2
|λ| = λλ
connects these two concepts (see page 69 for the definitions and the
basic properties of the absolute value and complex conjugate). For
n
z = (z 1 ,...,z n ) ∈ C , we define the norm of z by

2
2
z = |z 1 | +· · ·+|z n | .
The absolute values are needed because we want z to be a nonnega-
tive number. Note that

2
z = z 1 z 1 +· · ·+ z n z n .
2
We want to think of z as the inner product of z with itself, as we
n
did in R . The equation above thus suggests that the inner product of
n
w = (w 1 ,...,w n ) ∈ C with z should equal
w 1 z 1 +· · ·+ w n z n .

If the roles of the w and z were interchanged, the expression above
would be replaced with its complex conjugate. In other words, we
should expect that the inner product of w with z equals the complex
conjugate of the inner product of z with w. With that motivation, we
are now ready to define an inner product on V, which may be a real or
a complex vector space.
An inner product on V is a function that takes each ordered pair
(u, v) of elements of V to a number u, v à F and has the following
properties: