Let $E_i$ be a normed $\mathbb R$-vector space and $x_i,y_i\in E_i$ be linearly independent for $i\in\{1,2,3\}$. How can we show that $x_1\otimes x_2\otimes y_3$, $x_1\otimes y_2\otimes x_3$ and $y_1\otimes x_2\otimes x_3$ are linearly independent and $$u:=x_1\otimes x_2\otimes y_3+x_1\otimes y_2\otimes x_3+y_1\otimes x_2\otimes x_3$$ is not $0$?
Assume $$\lambda_1x_1\otimes x_2\otimes y_3+\lambda_2x_1\otimes y_2\otimes x_3+\lambda_3y_1\otimes x_2\otimes x_3\tag1=0.$$ We may write $$x_1\otimes(\lambda_1x_2\otimes y_3+\lambda_2y_2\otimes x_3)+\lambda_3y_1\otimes x_2\otimes x_3=0\tag2,$$ but I don't see how this could help to show that $\lambda_i=0$.
I know that $u$ is $0$ if and only if \begin{equation}\begin{split}&\varphi_1(x_1)\varphi_2(x_2)\varphi_3(y_3)\\&\;\;\;\;\;\;\;\;\;\;\;\;+\varphi_1(x_1)\varphi_2(y_2)\varphi_3(x_3)\\&\;\;\;\;\;\;\;\;\;\;\;\;+\varphi_1(y_1)\varphi_2(x_2)\varphi_3(x_3)=0\end{split}\tag3\end{equation} for all $\varphi_i\in E_i^\ast$.
EDIT: If $E_i$ is a $\mathbb R$-vector space, I'm defining $$(x_1\otimes x_2)(B):=B(x_1,x_2)\;\;\;\text{for }B\in\mathcal B(E_1\times E_2)\text{ and }x_i\in E_i,$$ where $\mathcal B(E_1\times E_2)$ is the space of bilinear forms on $E_1\times E_2$, and $$E_1\otimes E_2:=\operatorname{span}\{x_1\otimes x_2:E_i\in E_i\}\subseteq{\mathcal B(E_1\times E_2)}^\ast.$$