Inner Product Spaces, Hilbert Spaces

Abstract

The previous chapter introduced the concept of the norm of a vector as a generalization of the idea of the length of a vector. However, the length of a vector in ℝ² or ℝ³ is not the only geometric concept which can be expressed algebraically. If x = (x ₁, x ₂, x ₃) and y = (y ₁, y ₂, y ₃) are vectors in ℝ³ then the angle, θ, between them can be obtained using the scalar product (x, y) = x ₁ y ₁ + x ₂ y ₂ + x ₃ y ₃ = ∥x∥ ∥y∥ cos θ, where $\left\| x \right\| = \sqrt {x_1^2 + x_2^2 + x_3^2} = \sqrt {\left( {x,x} \right)}$ and $\left\| y \right\| = \sqrt {\left( {y,y} \right)}$ are the lengths of x and y respectively. The scalar product is such a useful concept that we would like to extend it to other spaces. To do this we look for a set of axioms which are satisfied by the scalar product in ℝ³ and which can be used as the basis of a definition in a more general context. It will be seen that it is necessary to distinguish between real and complex spaces.