The concept of an inner product is necessary before one can talk about orthogonal bases for vector spaces. Recall from elementary linear algebra that orthogonal bases were important in representing vectors. From a computational standpoint, as mentioned earlier, orthogonal bases can have a simplifying effect on certain types of approximation problem (e.g., least-squares approximations), and represent a means of controlling numerical errors due to so-called ill-conditioned problems.

Following our axiomatic approach, consider the following definition.

Definition 1.5: Inner Product Space, Inner Product An inner product space is a vector space X with an inner product defined on it. The inner product is a mapping ·, ·

|X × X → K that satisfies the following axioms: