11.1 General Theory
It is assumed here that the field of scalars is either ℝ or ℂ. The usual example of an inner product
space is ℂn or ℝn as described earlier. However, there are many other inner product spaces and
the topic is of such importance that it seems appropriate to discuss the general theory of these
Definition 11.1.1 A vector space X is said to be a normed linear space if there exists a
function, denoted by
) which satisfies the following axioms.
≥ 0 for all x ∈ X, and = 0
if and only if x = 0.
for all a ∈ F.
is called a norm.
is also often used. Not all norms are created equal. There are many
geometric properties which they may or may not possess. There is also a concept called an inner
product which is discussed next. It turns out that the best norms come from an inner
Definition 11.1.2 A mapping
V × V → F is called an inner product if it satisfies
the following axioms.
≥ 0 for all x ∈ V and equals zero if and only if x = 0.
whenever a,b ∈ F.
Note that 2 and 3 imply (x,ay + bz) = a(x,y) + b(x,z).
Then a norm is given by
It remains to verify this really is a norm.
Definition 11.1.3 A normed linear space in which the norm comes from an inner product
as just described is called an inner product space.
Example 11.1.4 Let V = ℂn with the inner product given by
This is an example of a complex inner product space already discussed.
Example 11.1.5 Let V = ℝn,
This is an example of a real inner product space.
Example 11.1.6 Let V be any finite dimensional vector space and let
be a basis.
and define the inner product by
The above is well defined because
is a basis. Thus the components
with any given x ∈ V
are uniquely determined.
This example shows there is no loss of generality when studying finite dimensional
vector spaces with field of scalars ℝ or ℂ in assuming the vector space is actually an
inner product space. The following theorem was presented earlier with slightly different
Theorem 11.1.7 (Cauchy Schwarz) In any inner product space
Proof: Let ω ∈ ℂ,|ω| = 1, and ω(x,y) = |(x,y)| = Re(x,yω). Let
Then from the axioms of the inner product,
then the inequality requires that |
= 0 since otherwise, you could pick large
and contradict the inequality. If
it follows from the quadratic formula
Earlier it was claimed that the inner product defines a norm. In this next proposition this
claim is proved.
Proposition 11.1.8 For an inner product space,
1∕2 does specify a norm.
Proof: All the axioms are obvious except the triangle inequality. To verify this,
The best norms of all are those which come from an inner product because of the following
identity which is known as the parallelogram identity.
Proposition 11.1.9 If
is an inner product space then for
1∕2, the following
It turns out that the validity of this identity is equivalent to the existence of an inner product
which determines the norm as described above. These sorts of considerations are topics for more
advanced courses on functional analysis.
Definition 11.1.10 A basis for an inner product space,
is an orthonormal basis
Note that if a list of vectors satisfies the above condition for being an orthonormal set, then
the list of vectors is automatically linearly independent. To see this, suppose
Then taking the inner product of both sides with uk,