Problem Session

Inner Product Spaces

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:

  1. <f, g> = <g, f>
  2. <f+g, h> = <f, h> + <g, h>
  3. <cf, g> = c<f, g>
  4. <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.

Group/Class Exercises

Section 5.5 - 2, 4, 5, 14, 24 (a and b only)

Please review Chapters 1-7 for class on Thursday.