Page 233 - A Course in Linear Algebra with Applications

7.2: Inner Product Spaces 217

inner product is to be defined on an arbitrary complex vector
space, there will have to be a change in the definition of the
inner product.
Let V be a vector space over C. An inner product on V
is a rule that assigns to each pair of vectors u and v i n F a
complex number < u, v > such that the following rules hold:
(i) < v, v > > 0 and < v, v > = 0 if and only if v = 0;
(ii) < u,v > = < v,u >;
(iii) < cu + dv, w > = c < u, w > +d < v, w > .

These are to hold for all vectors u, v, w and all complex scalars
c, d. Observe that property (ii) implies that < v,v > is
real: for this complex number equals its complex conjugate. A
complex vector space which is equipped with an inner product
is called a complex inner product space.
Our prime example of a complex inner product space is
n
C with the complex scalar product < X, Y > = X*Y. To
see that this is a complex inner product, we need to note that
X*Y = F*X and

< cX + dY, Z > = (cX + dY)*Z = cX*Z + dY*Z,

which is just c < X, Z > + d < Y, Z > .
Provided that the changes implied by the altered condi-
tions (ii) and (iii) are made, the concepts and results already
established for real inner product spaces can be extended to
complex inner product spaces. In addition, results stated for
real inner product spaces in the remainder of this section hold
for complex inner product spaces, again with the appropriate
changes.
Orthogonal complements
We return to the study of orthogonality in real inner prod-
uct spaces. We wish to introduce the important notion of the
orthogonal complement of a subspace. Here what we have in