An inner product in a linear space V is a rule that assigns a real scalar (denoted by <f, g>) to any pair f, g of elements of V, such that the following properties hold for all f, g, h in V and all real numbers c:
- <f, g> = <g, f>
- <f+g, h> = <f, h> + <g, h>
- <cf, g> = c<f, g>
- <f, f> > 0 for all nonzero f in V
A linear space endowed with an inner product is called an inner product space.
The norm (or magnitude) of an element f of an inner product space is: || f || = sqrt (<f, f>).
Two elements f and g of an inner product space are called orthogonal (or perpendicular) if <f, g> = 0.
Section 5.5 - 2, 4, 5, 14, 24 (a and b only)