Exercise 2, § 5

Let I and J be two finite sets, I_0 = I ∪ {α}, J_0 = J ∪ {β}, where α and β are two elements not belonging to I and J respectively, and suppose that I_0 and J_0 are disjoint. Let E be a vector space of finite rank over a commutative field K and let z be an element of T_J^I(E). For all i ∈ I, let W_i' be the subspace of E* consisting of the x* ∈ E* such that c_β^i(z ⊗ x*) = 0 (c_β^i being the contraction of index i ∈ I and index β ∈ J_0 in T_{J_0}^I(E)) and let V_i be the subspace of E orthogonal to W_i'. For all j ∈ J, let W_j be the subspace of E consisting of the x ∈ E such that c_j^α(z ⊗ x) = 0 (c_j^α being the contraction of index α ∈ I_0 and index j ∈ J in T_{J_0}^{I_0}(E)) and let V_j' be the subspace of E* orthogonal to W_j. Show that z belongs to the tensor product $\left( \bigotimes_{i \in I} V_i \right) \otimes \left( \bigotimes_{j \in J} V_j' \right)$ canonically identified with a subspace of T_J^I(E). Moreover, if (U_i)_{i \in I} is a family of subspaces of E, (U_j')_{j \in J} a family of subspaces of E* such that z belongs to the tensor product $\left( \bigotimes_{i \in I} U_i \right) \otimes \left( \bigotimes_{j \in J} U_j' \right)$, identified with a subspace of T_J^I(E), then V_i ⊂ U_i and V_j' ⊂ U_j' for all i ∈ I and j ∈ J (cf. II, § 7, no. 8, Proposition 17).