What are the sufficient and necessary conditions for representing every linear map $\varphi \in L(V_1\otimes\cdots\otimes V_k; W_1\otimes\cdots\otimes W_k)$ by a tensor in $L(V_1;W_1) \otimes\cdots\otimes L(V_k;W_k)$ that acts as follows $$ A_1 \otimes\cdots\otimes A_k: v_1 \otimes\cdots\otimes v_k \mapsto (A_1 v_1) \otimes\cdots\otimes (A_k v_k) $$ where $A_j \in L(V_j;W_j),\ j \in\{1,\dots,k\}$.
This question is exactly my question but the answer there doesn't provide details on how the inclusion $$ L(V_1;W_1) \otimes\cdots\otimes L(V_k;W_k) \hookrightarrow L(V_1\otimes\cdots\otimes V_k; W_1\otimes\cdots\otimes W_k) $$ fails to be an isomorphism for infinite dimesnional vector spaces. Could someone explain that by an example ?
The standard example is to take $$ V_1 = V, W_1 = \mathbb{F}, V_2 = \mathbb{F}, W_2 = V $$ for some $\mathbb{F}$ vector space $V$. Then
$$ L(V_1, W_1) = L(V,\mathbb{F}) = V^{*}, \,\,\, L(V_2,W_2) = L(\mathbb{F}, V) \cong V $$ and we have an inclusion $$ L(V_1, W_1) \otimes L(V_2, W_2) \cong V^{*} \otimes V \hookrightarrow L(V \otimes \mathbb{F}, \mathbb{F} \otimes V) \cong L(V, V) $$ given by $\varphi \otimes v \colon w \mapsto \varphi(w)v. $ Note that an element of $V^{*} \otimes V$ is a finite linear combination of simple tensors $\varphi \otimes v$ which means that in particular, the linear map that it defines on $V$ must be of finite rank. So any linear map on $V$ which is not of finite rank (such as the identity) cannot be in the image of the inclusion.