Inner Product Spaces

Abstract

So far in our study of vector spaces and linear transformations we have made no use of the notions of length and angle, although these concepts play an important role in our intuition for the vector algebra of ℝ²and ℝ³. In fact, the length of a vector and the angle between two vectors play very important parts in the further development of linear algebra, and it is now appropriate to introduce these ingredients into our study. There are many ways to do this, and in the approach that we will follow both length and angle will be derived from a more fundamental concept called a scalar or inner product of two vectors. Quite likely the student has encountered the scalar product in the guise of the dot product of two vectors in ℝ³, which is usually defined by the equation

$$ A \cdot B = \left| A \right|\left| B \right|\cos \left( \vartheta \right) $$

where |A| is the length of the vector A, similarly for B, and ϑ is the angle between A and B. In the study of vectors in IR³ this is a reasonable way to introduce the scalar product, because lengths and angles are already defined and wellstudied concepts of geometry. In a more abstract study of linear algebra such as we are undertaking, this is not possible, for what is the length of a polynomial (vector) in P₄(IR)? Instead, we will again resort to the use of the axiomatic method. Having introduced vector spaces by axioms, it is not at all unreasonable to employ additional axioms to impose further structure on them.