It is assumed here that the field of scalars is either ℝ or ℂ. The usual example of an inner product space is ℂ^{n} or ℝ^{n} 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 spaces.
Definition 11.1.1 A vector space X is said to be a normed linear space if there exists a function, denoted by
This function
The notation
Definition 11.1.2 A mapping
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 = ℂ^{n} with the inner product given by

This is an example of a complex inner product space already discussed.
Example 11.1.6 Let V be any finite dimensional vector space and let

and define the inner product by

where

The above is well defined because
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 notation.
Proof: Let ω ∈ ℂ,ω = 1, and ω(x,y) = (x,y) = Re(x,yω). Let

Then from the axioms of the inner product,

This yields

If

Earlier it was claimed that the inner product defines a norm. In this next proposition this claim is proved.
Proof: All the axioms are obvious except the triangle inequality. To verify this,
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.
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 u_{k},
