Exercise 12.3 (Tensor product is bilinear and associative)

Proof. Pick an arbitrary argument of $F\otimes G$. It is either in the index $1\le I\le k$ of $F$ or $1\le j\le l$ of $W$. In the former case, let $a,a^{\prime }$ be scalars and let $v_i,v_i^{\prime }\in V_i$. Then,

$$\begin{array}{llll}\hfill & F\otimes G(v_1,\dots ,a{v}_i+a^{\prime }{v}_i^{\prime },\dots ,v_k,w_1,\dots ,w_l)& \hfill & \\ \hfill & =F(v_1,\dots ,a{v}_i+a^{\prime }{v}_i^{\prime },\dots ,v_k)\times G(w_1,\dots ,w_l)& \hfill & \\ \hfill & =\left[\mathit{aF}(v_1,\dots ,v_i,\dots ,v_k)+a^{\prime }F(v_1,\dots ,v_i^{\prime },\dots ,v_k)\right]\times G(w_1,\dots ,w_l)& \hfill & \\ \hfill & =\mathit{aF}(v_1,\dots ,v_i,\dots ,v_k)\times G(w_1,\dots ,w_l)+a^{\prime }F(v_1,\dots ,v_i^{\prime },\dots ,v_k)\times G(w_1,\dots ,w_l)& \hfill & \\ \hfill & =a(F\otimes G)(v_1,\dots ,v_i,\dots ,v_k,w_1,\dots ,w_l)+a^{\prime }(F\otimes G)(v_1,\dots ,v_i^{\prime },\dots ,v_k,w_1,\dots ,w_l).& \hfill & \end{array}$$

The case for $w_j,w_j^{\prime }\in W_j$ follows analogously. □

Proof. The following trivial computation breaks down to using the associativity of product on the domain space $ℝ$. For any $v_1\in V,\dots ,v_k\in V_k$, $w_1\in W_1,\dots ,w_l\in W_l$ and $u_1\in U_1,\dots ,u_l\in U_l$, we have:

$$\begin{array}{llllll}\hfill & ((F\otimes G)\otimes H)((v_1,\dots ,v_k,w_1,\dots ,w_k),u_1,\dots ,u_p)& \hfill & & \hfill & \\ \hfill & =(F\otimes G)(v_1,\dots ,v_k,w_1,\dots ,w_k)\times H(u_1,\dots ,u_p)& \hfill & & \hfill & \\ \hfill & =\left(F(v_1,\dots ,v_k)\times G(w_1,\dots ,w_k)\right)\times H(u_1,\dots ,u_p)& \hfill & & \hfill & \\ \hfill & =F(v_1,\dots ,v_k)\times \left(G(w_1,\dots ,w_k)\times H(u_1,\dots ,u_p)\right)& \hfill \text{associativity of }\text{×}\text{ on }ℝ& & \hfill & & \hfill \\ \hfill & =F(v_1,\dots ,v_k)\times \left((G\otimes H)(w_1,\dots ,w_k,u_1,\dots ,u_p)\right)& \hfill & & \hfill & \\ \hfill & =(F\otimes (G\otimes H))((v_1,\dots ,v_k),w_1,\dots ,w_k,u_1,\dots ,u_p).& \hfill & & \hfill & \end{array}$$ □