Multilinear Algebra

Theorem 1.1. Suppose that V , W are finite dimensional vector spaces over a field F . Then there exists a vector space T over F , and a bilinear map φ : V ×W → T such that T satisfies the following “universal property”: If U is a vector space over F and g : V ×W → U is a bilinear map then there exists a unique linear map g∗ : T → U such that for all (v, w) ∈ V × W we have g(v, w) = g∗(φ(v, w)). Lemma 1.2. Suppose that T1, with bilinear map φ1 : V ×W → T1, satisfies the universal property of theorem 1.1, and T2, with bilinear map φ2 : V ×W → T2, satisfies the universal property of theorem 1.1. Then there exists a (unique) linear isomorphism ψ : T1 → T2 such that ψφ1 = φ2.