ABSTRACT
Defi nition 3.1 (Real Inner Product Vector Spaces). A real inner product vector space (V, +, .) (vector space, in short) is a real vector space together with a map V × V → R, (u, v) ↦ u.v, (3.1) called a real inner product, satisfying the following properties for all u, v, w ∈ V and r ∈ R: 1) v.v ≥ 0, with equality if, and only if, v = 0. 2) u.v = v.u 3) (u + v).w = u.w + v.w 4) (ru).v = r(u.v).
