Which ones of these two shows that the tensor product operation on two multilinear functions is bilinear?

Question

Suppose $v \in V_1 \times \cdots \times V_k$ and $w \in W_1\times \cdots \times W_l$. The tensor product of $F \in L(V_1 \times \cdots \times V_k, \mathbb R)$ and $G \in L(W_1 \times \cdots \times W_l, \mathbb R)$ is defined by $$F\otimes G(v,w)=F(v)G(w).$$

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If $F \in L(V_1 \times \cdots \times V_k, \mathbb R)$ and $G \in L(W_1 \times \cdots \times W_l, \mathbb R)$, how do we show that the tensor product operation is bilinear on $F$ and $G$?

Do we have to show that $$F\otimes G(av_1+bv_2, w)=aF\otimes G(v_1,w)+bF\otimes G(v_2,w)?$$

Or do we have to show that $$(aF_1+bF_2)\otimes G(v,w)=aF_1\otimes G(v,w)+bF_2\otimes G(v,w)?$$

Answer 1

To show that the tensor product operation is bilinear on $F$ and $G$, we must show:

$$(aF_1+F_2)\otimes G(v,w)=aF_1\otimes G(v,w)+F_2\otimes G(v,w),$$ $$F\otimes (aG_1 +G_2)(v,w)=F\otimes aG_1(v,w)+F\otimes G_2(v,w).$$

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Your first one: $F\otimes G(av_1+bv_2, w)=aF\otimes G(v_1,w)+bF\otimes G(v_2,w)$ does not show that the tensor product is bilinear on $F$ and $G$. It just shows that it is linear in the arguments of the tensor product, but we already know this since $F$ and $G$ are both multi-linear functions.