Exercise 3.E.4

In this solution, we are going to create an $S$ which maps from $L(V_1,W)\times \cdots \times L(V_m,W)\to L(V_1\times \cdots \times V_m,W)$

Define $T:V_1\times \cdots \times V_m\to W$ as

where $T_j\in L(V_j,W)$ for $j\in \{1,...,m\}$

Now define $S:L(V_1,W)\times \cdots \times L(V_m,W)\to L(V_1\times \cdots \times V_m,W)$ where $(T_1,...,T_m)$ is mapped to $T$

To prove $S$ is injective, let $l\in \text{null}S$ Then, $\forall <!--\; FUNCTION\; APPLICATION\; -->v_1,...,v_m$

Therefore

Therefore $\text{null}S=\{0\}$ To prove surjectivity, for any $K:V_1\times \cdots \times V_m\to W$, let

For $j\in \{1,...,m\}$ Then $T=T_1+\cdots +T_m$, proving surjectivity. □