Linear Functional Analysis pp 51-85 | Cite as

# 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 ℝ^{2} or ℝ^{3} 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 ℝ^{3} 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 ℝ^{3} 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.

### Keywords

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