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Properties of Inner Products

Module by: Mark A. Davenport. E-mail the author

Inner products and their induced norms have some very useful properties:

  • Cauchy-Schwartz Inequality: |〈x,y〉|≤xy|〈x,y〉|≤xy with equality iff ∃α∈C∃α∈C such that y=αxy=αx
  • Pythagorean Theorem: 〈 x , y 〉 = 0 ⇒ x + y 2 = x - y 2 = x 2 + y 2 〈 x , y 〉 = 0 ⇒ x + y 2 = x - y 2 = x 2 + y 2
  • Parallelogram Law: x + y 2 + x - y 2 = 2 x 2 + 2 y 2 x + y 2 + x - y 2 = 2 x 2 + 2 y 2
  • Polarization Identity: Re [ 〈 x , y 〉 ] = x + y 2 - x - y 2 4 Re [ 〈 x , y 〉 ] = x + y 2 - x - y 2 4

In R2R2 and R3R3, we are very familiar with the geometric notion of an angle between two vectors. For example, if x,y∈R2x,y∈R2, then from the law of cosines, 〈x,y〉=xycosθ〈x,y〉=xycosθ. This relationship depends only on norms and inner products, so it can easily be extended to any inner product space.

Figure 1
An illustration of a pair of vectors x and y that make an angle of theta.

Definition 1

The angleθθ between two vectors x,yx,y in an inner product space is defined by cosθ=〈x,y〉xycosθ=〈x,y〉xy

Definition 2

Vectors x,yx,y in an inner product space are said to be orthogonal if 〈x,y〉=0〈x,y〉=0.

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