Scalar Products and Orthogonality

Abstract

Let V be a vector space over a field K. A scalar product on V is an association which to any pair of elements v, w of V associates a scalar, denoted by <v, w>, or also v·w, satisfying the following properties:

• SP 1. We have <v, w> = <w, v> for all v, w ∈ V.

• SP 2. If, u, v, w are elements of V, then

  $$<u,v+ w>=<u,v>+<u,w>$$

  .

• SP 3. If x ∈ K, then

  $$<xu,v>= x<u,v>$$

  and

  $$u,xv>=x<u,v>$$