# Inner Product Spaces, Hilbert Spaces

• Bryan Patrick Rynne
• Martin Alexander Youngson
Part of the Springer Undergraduate Mathematics Series book series (SUMS)

## 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.

Sine Eosine